Stars and combs

In this subsection, we first recall the Star-Comb Lemma, which is featured in Chapter 4 of this dissertation and some of the relevant definitions.

Recall that we call an undirected graph astar if it is isomorphic to the complete bipartite graph K_1,κ for some cardinal κ, where the vertices of degree 1 are its leaves and the vertex of degree κ is its centre.

An undirected multigraph that does not contain a ray is calledrayless. We define the ends of a digraph D as the ends of Un(D). If ω is an end of an undirected multigraph G (resp. of a digraph D), we call any ray that is contained in ω an ω-ray.

Recall that acombC is an undirected graph that is the union of a rayRtogether with infinitely many disjoint undirected finite paths each of which has precisely one vertex in common with R, which has to be an endvertex of that path. The ray R is called the spine of C. The endvertices of the finite paths that are not onR together with the endvertices of the trivial paths are the teeth of C.

The following lemma, the Star-Comb Lemma, is a basic tool in infinite graph theory, cf. Lemma 4.2.3. We shall only apply it for vertex sets of cardinality ℵ₀ and ℵ₁ in this chapter.

Lemma 6.2.5. Let G be an infinite connected undirected multigraph and let U ⊆V(G) be such that |U|=κ for some regular cardinal κ. Then there exists a set U⁰ ⊆U with |U⁰|=|U| such that G either contains a comb whose set of teeth is U⁰ or a subdivided star whose set of leaves is U⁰.

Later in Section6.8 we shall need two lemmas about digraphs, which are similar to Lemma 6.2.5 for undirected infinite graphs. In order to state these lemmas we have to give some definitions:

We call a digraphS a subdivided out-star (resp. subdivided in-star) if Un(S) is a subdivided star, precisely one vertexcofS has in-degree 0 (resp. out-degree 0) and all other vertices ofS have in-degree 1 (resp. out-degree 1). We call the vertex c of S the centre of S. Furthermore, we call a vertex v of S aleaf of S if v is a leaf of Un(S). Note that the centre ofS coincides with the centre of Un(S).

Next we call a digraphC aweak forward (resp.weak backward)comb if Un(C) is a comb andCorients the spine of Un(C) such that it is a forwards (resp. backwards) directed ray. We call a vertex v of C a tooth of C if v is a tooth of Un(S). A weak forward comb C with set of teethT and spine S is called a forward out-comb (resp. forward in-comb) if each S–T path in Un(C) is a directed one from S toT (resp. from T to S) in C. Analogously, a weak backward comb C with set of

teeth T and spineS is called abackward out-comb (resp.backward in-comb) if each S–T path in Un(C) is a directed one from S toT (resp. from T toS) in C.

In a digraph D with a vertexv ∈V(D) letN_∞⁺(v) denote the set of all vertices ofDthat can be reached inDby some directed path starting atv. IfZ is a subgraph of D or some subset of the vertices ofD, we define N_∞⁺(Z) = ^S_z∈V(Z)N_∞⁺(z). The notation N_∞⁻(v) and N_∞⁻(Z) is analogously defined.

Let D be a digraph, v ∈V(D) and ω be an end of D. Now we call ω reachable from v if there exists an ω-rayR in D that is a forwards directed ray whose start vertex isv. Similarly, we call v reachable from ω if there exists an ω-ray R in D that is a backwards directed ray whose start vertex is v. Similarly as for vertices, we let N_∞⁺(ω) denote the set of all vertices inD that can be reached from ω and we denote by N_∞⁻(ω) the set of all vertices in D that reachω.

Recall that we call a digraph A anout-arborescence rooted in r∈V(A) if

• Un(A) is a tree;

• d⁻_A(v) = 1 for all v ∈V(D)r{r}; and

• d⁻_A(r) = 0.

A straight-forward transfinite construction yields the following remark:

Remark 6.2.6. Let D be a weakly connected digraph and U ⊆V(D). Let U ⊆N_∞⁺(v) for some vertex v of D. Then D contains an out-arborescence rooted in v that containsU.

Now we are able to state and prove two lemmas about digraphs, which have a certain resemblance with Lemma 6.2.5.

Lemma 6.2.7. Let D be a weakly connected digraph and U ⊆V(D) an infinite vertex set. If U ⊆N_∞⁺(v) (resp. U ⊆N_∞⁻(v)) holds for some vertex v of D, then there exists an infinite subset U⁰ ⊆U such that one of the following assertions is true:

(a) there exists a subdivided out-star (resp. subdivided in-star) in D whose set of leaves is U⁰; or

(b) there exists a forward out-comb (resp. backward in-comb) in D whose set of teeth is U⁰.

Proof. We assume that U ⊆N_∞⁺(v) holds for some vertexv of D. The case that U ⊆N_∞⁻(v) follows from the first case by reversing the orientation of each edge in D.

LetA⊆Dbe an out-arborescence rooted inv that containsU as in Remark6.2.6.

Applying Lemma 6.2.5 to Un(A) and U yields either subdivided star S with leaves U⁰ or a comb C with teeth U⁰ for some infinite U⁰ ⊆U.

In the first case, letS⁰ be the subdigraph of D such that Un(S⁰) = S. Without loss of generality we may assume that the centre cof S⁰ has in-degree 0 in S⁰. So since no vertex has in-degree 2, every path from a cto a leaf u is a directedc–u path. Hence S⁰ is the desired subdivided out star.

In the second case, letC⁰ be the subdigraph of Dsuch that Un(C⁰) =C. Note that the spine R of C⁰ contains at most one vertex w with d⁺_R(w) = 2, since otherwise it would contain a vertex with in-degree 2 as well. Hence as before, we may assume without loss of generality that the spine R of C⁰ contains no such vertexw withd⁺_R(w) = 2. And as before, every path from the spine to a tooth uof the comb is a directed R–upath. Hence C⁰ is the desired forward out-comb.

In contrast to Lemma 6.2.7whose statement contains an assumption about the reachability of an infinite vertex set from some vertex, the statement of the next lemma has a similar assumption, but about the reachability from some end.

Lemma 6.2.8. Let D be a weakly connected digraph and U ⊆V(D) an infinite vertex set. If U ⊆N_∞⁺(ω) (resp. U ⊆N_∞⁻(ω)) holds for some end ω of D, then there exists an infinite subset U⁰ ⊆U such that one of the following assertions is true:

(a) there exists a subdivided out-star (resp. subdivided in-star) in D whose set of leaves is U⁰; or

(b) there exists a forward out-comb (resp. backward in-comb) in D whose set of teeth is U⁰.

(c) there exists a weak backward (resp. forward) comb in D whose set of teeth is U⁰.

Proof. We only give a proof for the case that U ⊆N_∞⁺(ω) holds for some end ω of D since the case that U ⊆N_∞⁻(ω) follows from the first statement by reversing all edges in D.

Let ω be an end of D as in the statement. Suppose first that there exists a backwards directed ω-ray R in D such that an infinite set U₁ ⊆U exists with the property that U₁ ⊆N_∞⁺(R). If there already exists a finite segment I of R such that U₂ ⊆N_∞⁺(I) for some infinite setU₂ ⊆U₁, we are done by Lemma 6.2.7.

Hence, we may assume that every finite segment of R reaches only finitely many vertices of U₁. Using this property it is a standard task to recursively define along R a backward out-comb. We omit this definition here. This completes the proof for the first case.

Now let us consider the remaining case where U ∩N_∞⁺(R) is finite for each backwards directed ω-ray R. For every u∈U let R_u denote some backwards directed ω-ray whose start vertex is u. Note that if R_u∩R_v 6=∅ for u, v ∈U, then {u, v} ⊆N_∞⁺(R_u)∩N_∞⁺(R_v). Hence, for each u∈U there are only finitely many v ∈U such that Ru∩Rv 6=∅. Using this observation we can recursively define an infinite set U₂ ⊆U such that R_u∩R_v =∅ holds for all u, v ∈U₂ with u6=v. Since each ray R_u for u∈U₂ is an ω-ray, there are, for any two distinct vertices u, v of U₂, infinitely many disjoint undirected paths between R_u andR_v

in D. Using this property, it is again a standard task to recursively define a weak backward comb whose spine is any previously chosen R_u for u∈U₂.