Theorem

All graphs in this chapter are simple and undirected.

For a finite set M of vertices of a graph G and an end ω of G, let C(M, ω) denote the unique component of G−M that contains a tail of every ω-ray.

For the proof of Theorem 3.1.1 we shall use the following characterisation of ω-rays.

Lemma 3.2.2. Let G be a graph, ω an end of G and R_max an inclusion-maximal set of pairwise disjoint ω-rays. A ray R ⊆G is an ω-ray, if and only if it meets rays of R_max infinitely often.

Proof. LetW denote the set ^S{V(R)|R ∈ R_max}.

If R is an ω-ray, then each tail of R meets a ray of R_max since R_max is inclusion-maximal. Hence R meets W infinitely often.

Suppose for a contradiction that R is an ω⁰-ray for an end ω⁰ 6=ω of G that contains infinitely many vertices of W. LetM be a finite set of vertices such that the two componentsC :=C(M, ω) andC⁰ :=C(M, ω⁰) ofG−M are different. By the pigeonhole principle there is either oneω-ray ofR_maxcontaining infinitely many vertices of V(C⁰)∩V(R)∩W, or infinitely many disjoint rays ofR_max containing those vertices. In both cases we get an ω-ray with a tail in C⁰, since we cannot leaveC⁰ infinitely often through the finite setM. But this contradicts the definition of C.

A natural strategy for constructing up to infinitely many disjoint rays is to inductively construct them in countably many steps. In each step we fix only finitely many finite paths as initial segments instead of whole rays, while extending previously fixed initial segments and ensuring that they can be extended to rays.

This strategy is for example used by Halin [30, Satz 1] to prove that the maximum number of disjoint rays in an end is well-defined. Our proof of Theorem 3.1.1 is also based on that strategy. In order to guarantee that the set of rays we construct turns out to devour the end, we also fix an inclusion-maximal set of vertex disjoint rays of our specific end, so a countable set, and an enumeration of the vertices on these rays. Then we try in each step to either contain or separate the least vertex with respect to the enumeration that is not already dealt with from the end with appropriately chosen initial segments if possible. Otherwise, we extend a

finite number of initial segments while still ensuring that all initial segments can be extended to rays. Although it is impossible to always contain or separate the considered vertex with our initial segments while being able to continue with the construction, it will turn out that the rays we obtain as the union of all initial segments actually do this.

For a vertexv and an end ω of a graphG we say that a vertex set X ⊆V(G) separates v from ω if there does not exist any ω-ray that is disjoint from X and contains v.

Furthermore, in addition to the notations for paths introduced in the beginning of this theses, for Q being a v–w path we writevQ¯ for the path that is obtained from Qby deleting w.

Proof of Theorem 3.1.1. Let us fix a finite or infinite enumeration {R_j|j <|R|}

of the rays inR. Furthermore, lets_j denote the start vertex ofR_j for everyj <|R|

and define S :={s_j|j <|R|}.

Next we fix an inclusion-maximal set Rmax of pairwise disjoint ω-rays and an enumeration {v_i|i∈N} of the vertices in W :=^S{V(R)|R∈ R_max}. This is possible since ω is countable by assumption.

We do an inductive construction such that the following holds for everyi∈N: (1) P_sⁱ is a path with start vertexs for every s ∈S.

(2) P_sⁱ =s for all but finitely many s∈S.

(3) P_sⁱ⁻¹ ⊆P_sⁱ for every s∈S.

(4) For every s=s_j ∈S with j <min{i,|R|} there is a wⁱ_s∈W ∩(P_sⁱrP_sⁱ⁻¹).

(5) P_sⁱ and P_sⁱ0 are disjoint for all s, s⁰ ∈S with s6=s⁰.

(6) For everys ∈S there exists an ω-ray Rⁱ_s with P_sⁱ as initial segment and s as start vertex such that all raysRⁱ_s are pairwise disjoint.

If possible and not spoiling any of the properties (1) to (6), we incorporate the following property:

(∗) ^S

s∈S

P_sⁱ either contains v_i−1 or separatesv_i−1 fromω if i >0.

We begin the construction fori= 0 by defining P_s⁰ :=s=:P_s⁻¹ for every s ∈S.

All conditions are fulfilled as witnessed by R⁰_sj :=R_j for every j <|R|.

Now suppose we have done the construction up to some numberi∈N. If we can continue with the construction in step i+ 1 such that properties (1) to (6) together with (∗) hold, we do so and define all initial segmentsP_sⁱ⁺¹ and raysR_sⁱ⁺¹ accordingly. Otherwise, we set for all s∈S

P_sⁱ⁺¹ :=sRⁱ_swⁱ_s if s=s_j for j <min{i+ 1,|R|} and P_sⁱ⁺¹ :=P_sⁱ otherwise,

where wⁱ_s denotes the first vertex of W on Rⁱ_srP_sⁱ which exist by Lemma 3.2.2.

With these definitions properties (1) up to (5) hold for i+ 1. Witnessed by R_sⁱ⁺¹ :=Rⁱ_s for every s∈S we immediately satisfy (6) too. This completes the inductive part of the construction.

Using the pathsP_sⁱ we now define the desiredω-rays ofR⁰. We setR⁰_s :=^S_i∈NP_sⁱ for every s∈S andR⁰ :={R⁰_s|s∈S}. Properties (1), (3)and (4) ensure that R⁰_s is a ray with start vertex s for every s∈S, while we obtain due to property (5) that all rays R_s⁰ are pairwise disjoint. Property(4) also ensures that all rays in R⁰ are ω-rays by Lemma 3.2.2.

It remains to prove that the setR⁰devours the endω. Suppose for a contradiction that there exists anω-ray R disjoint from ^SR⁰. By the maximality of our chosen set of ω-rays R_max, we know that R contains a vertex v_j for some j ∈N. In the next paragraph we will show how we could have proceeded in step j+ 1 to incorporate property (∗) as well. For an easier understanding of the technical definitions of that paragraph we refer to Figure 3.2.1.

Without loss of generality, let v_j be the start vertex of R. Let P be an R–

SR⁰ path among those ones that are disjoint from ^S_s∈SsP¯_s^j+1 for which v_jRp is as short as possible where p denotes the common vertex of P and R. Such a path exists, because all rays in R⁰∪ {R} are equivalent and ^S_s∈SsP¯_s^j+1 is finite by property (2). Lett ∈S and q∈V(G) be such that V(P)∩V(R⁰_t) ={q}.

Furthermore, let R^∗ be an ω-ray with start vertex r^∗ ∈R such that R^∗ is disjoint from ^S_s∈SR⁰_s and P(pRr^∗)∩R^∗ ={r^∗} for which vjRr^∗ is as short as possible.

Since pandpR are candidates forr^∗ andR^∗, respectively, such a choice is possible.

We define

Pˆ_t^j+1 := (tR_t⁰q)P(pRr^∗) and Rˆ^j+1_t := ˆP_t^j+1R^∗ ;

and replace in step j+ 1 the ray R^j+1_t by ˆR^j+1_t , the path P_t^j+1 by ˆP_t^j+1 and for all s∈Sr{t} the rayR^j+1_s by R⁰_s while keeping P_s^j+1 as it was defined. By this construction properties (1) to (6) are satisfied.

vj

r^∗ p R

P

q

S3s t

R⁰3R_s⁰

P_s^j+1 P_t^j+1

R^∗ R⁰_t

Figure 3.2.1.: Sketch of the situation above. The rays in R⁰ are drawn vertically, with their fixed initial segments from step j+ 1. Horizontally drawn is the rayRthat is supposed to contradict thatR⁰ devoursωwith its start vertex v_j ∈W. The R–^SR⁰ path P is chosen with its vertex p on R as close to v_j as possible. The rayR^∗ is chosen disjoint to the rays inR⁰ and except from its start vertex r^∗ on R disjoint from the initial segment of R upto p again with r^∗ as close tov_j as possible.

The ray ˆR^j+1_t is highlighted in grey with its initial segment fixed up to r^∗.

Now we show that (∗) holds as well. Suppose for a contradiction that there exists an ω-rayZ disjoint from^S_s∈Sr{t}P_s^j+1∪Pˆ_t^j+1 with start vertexv_j. First note that Z is disjoint from r^∗Rp⊆Pˆ_t^j+1. Let us now show that Z is also disjoint from pR∪^S_s∈SR⁰_s. Otherwise, let z denote the first vertex along Z that lies inpR∪^S_s∈SR⁰_s. However,z cannot be contained in pR, as this would contradict the choice of r^∗, and it cannot be an element of^S_s∈SR⁰_ssince this would contradict the choice ofp. But now withZ being not only disjoint frompR∪^S_s∈SR⁰_sbut also from r^∗Rp, we get again a contradiction to the choice of r^∗. Hence, we would have been able to incorporate property (∗) without violating any of the properties(1) to (6) in step j+ 1 of our construction. This, however, is a contradiction since we

always incorporated property(∗)under the condition of maintaining properties (1) to(6). So we arrived at a contradiction to the existence of the ray R since by (∗) every ray containingv_j meets the initial segments of rays fixed in our construction at step j+ 1. Therefore, the set R⁰ devours the end ω.