Enlarging vertex-flames in countable digraphs

A rooted digraph is a vertex-flame if for every vertex $v$ there is a set of internally disjoint directed paths from the root to $v$ whose set of terminal edges covers all ingoing edges of $v$. It was shown by Lov\'{a}sz that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by the third author and the analogue of Lov\'{a}sz' result for countable digraphs was shown. We strengthen this result by proving that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame.


Introduction
Given a rooted digraph D with root r the local connectivity from r to v, written κ D (r, v), is the size of the largest set of internally disjoint paths from r to v in D. Lovász showed in [7] that there always exists a spanning subgraph which preserves all the local connectivities from r, whilst keeping only κ D (r, v) ingoing edges at each v.
More precisely, we say that a rooted digraph is a vertex-flame (or shortly flame) if for every vertex v (other than the root) there is a set of internally disjoint directed paths from the root to v whose set of terminal edges is in D (v) (i.e., the set of ingoing edges of v).
Lovász showed that every finite rooted digraph D contains a vertex-flame which is large, i.e., it has the same local connectivity from the root to any other vertex as in D. 1 The second author was supported by the Institute for Basic Science (IBS-R029-C1).
The third author would like to thank the generous support of the Alexander von Humboldt Foundation and NKFIH OTKA-129211. 1 Edge-flames can be defined in an analogous manner by replacing 'internally disjoint' with 'edge-disjoint'.
In this case, preservation of local edge-connectivities from the root in an edge-flame can be accomplished.
In fact Lovász originally proved the edge variant, from which the vertex version can be deduced as a corollary. Theorem 1.1 (Lovász). If D is a finite digraph with r ∈ V (D), then there is a spanning subgraph F of D such that for every v ∈ V (D) − r, Theorem 1.1 above was rediscovered by Calvillo-Vives in [2] in a stronger form: Theorem 1.2 (Calvillo-Vives). Any vertex-flame in a given finite rooted digraph can be extended to a large one.
Since in particular the spanning subgraph with empty edge set is a flame, this clearly implies Theorem 1.1.
Recently, the third author proved a generalisation of Theorem 1.1 for countably infinite digraphs [5]. As it is common in the context of infinite graphs, a naive cardinality-based approach yields a much weaker result than a more structural generalisation of the problem.
For example, Erdős conjectured a structural generalisation of Menger's theorem for infinite graphs, which was eventually proved in an influential paper of Aharoni and Berger [1]. In particular they showed that in every digraph D for every x = y ∈ V (D) with xy / ∈ E(D) there is an orthogonal pair of a set P of internally disjoint directed paths from x to y and an S ⊆ V (D) \ {x, y} separating y from x (i.e., meeting every path from x to y), where orthogonality means that S contains exactly one vertex from each path in P, and no other vertices.
Motivated by this, the third author gave a structural characterisation of when it can be said that a subgraph preserves the local connectivity from the root to each vertex in an infinite digraph in the spirit of the Aharoni-Berger theorem. More precisely we say that a spanning subdigraph L of a rooted digraph D is D-large (or large) if for every v = r the digraphs D − rv and L − rv share such an orthogonal pair for r and v and furthermore rv ∈ E(L) if rv ∈ E(D). For more details on the definition, see Section 2.
With this generalisation, the third author showed the following. The structure of this paper is as follows. In Section 2 we give some basic definitions and citations for some theorems that we use. In particular in Section 2.4 we will introduce the notion of a G-quasi-flame and state a key lemma, using which, in Section 3, we will prove Theorem 1.4. In Section 4 we will introduce the concept of incompressibility and develop the tools that we need to prove the key lemma in Section 5.
In this paper, D will denote a (usually infinite) digraph with vertex set V (D) and edge set E(D). For the questions we are considering we may assume without loss of generality that D does not contain any parallel edges or loops.
We denote an edge e of D directed from v to w also by the string vw. Here, v denotes the tail of e and w denotes the head of e. Often we will consider D as a rooted digraph, i.e., a digraph with one of its vertices r marked as a root. For simplicity we may always assume that the root has no ingoing edges, since such edges are not contained in any directed path starting from the root, and hence they are neither contained in any flame nor relevant for the notion of largeness. Since the root will be the same vertex r in the whole paper, we omit it from our notation without risking any confusion.
Given a vertex v ∈ V (D), we denote by in D (v) and out D (v) the set of ingoing and outgoing edges of v respectively. Here, and for other notation given in this section, we introduce the subscript since we will often need to work inside subdigraphs of D.
For a vertex or edge x we denote by D − x the digraph obtained from D by deleting x.
Similarly for a set X of either vertices or edges we denote by D − X the digraph obtained from D by deleting X. Lastly, given a set X and an element x ∈ X, we denote by X − x the set X \ {x}.
Let v ∈ V (D) and I ⊆ in D (v). We define the restriction of D to I at v as the digraph D − (in D (v) \ I) and denote it by D ↾ v I.

Path-systems.
Let x, y ∈ V (D) be distinct and let X, Y ⊆ V (D). We introduce the following notation for paths.
• An (x, y)-path is a directed path with initial vertex x and terminal vertex y. We call the (x, y)-path consisting of a single edge xy trivial. • An (X, Y )-path is a directed path whose initial vertex is contained in X, whose terminal vertex is contained in Y and which is internally disjoint from X ∪ Y .
• An (x, Y )-path is a directed path whose initial vertex is x, whose terminal vertex is contained in Y and which is internally disjoint from Y .
• An (X, y)-path is a directed path whose initial vertex is contained in S, whose terminal vertex is y and which is internally disjoint from X.
We also introduce the following notation for sets of paths.
• An (x, y)-path-system is a set of (x, y)-paths, which are pairwise internally disjoint.
• An (X, Y )-path-system is a set of (X, Y )-paths, which are pairwise disjoint.
• An (x, Y )-path-system is a set of (x, Y )-paths, which are pairwise disjoint but for x.
• An (X, y)-path-system is a set of (X, y)-paths, which are pairwise disjoint but for y.
For a set P of paths, we write V − (P) and E − (P) for the set of the initial vertices, and initial edges, respectively, of the paths in P. Similarly, we write V + (P) and E + (P) for the set of the terminal vertices, and terminal edges, respectively, of the paths in P.
• An (X, Y )-separation is a set S ⊆ V (D) of vertices that meets every (X, Y )-path.
A set of paths P and a vertex set S, are orthogonal to each other if |V (P ) ∩ S| = 1 for all P ∈ P and S ⊆ V (P). We write P ⊥ S if P and S are orthogonal.
• Let P be an (x, y)-path-system. We call P an Erdős-Menger (x, y)-path-system if there exists an (x, y)-separation S with P ⊥ S. Similarily, an (x, y)-separation S is called an Erdős-Menger (x, y)-separation if there is an (x, y)-path-system P with P ⊥ S. We denote the set of Erdős-Menger (x, y)-path-systems by P D (x, y) and the set of Erdős-Menger (x, y)-separations by S D (x, y). • Let P be an (X, Y )-path-system. We call P an Erdős-Menger (X, Y )-path-system if there exists an (X, Y )-separation S with P ⊥ S. An (X, Y )-separation S is called an Erdős-Menger (X, Y )-separation if there is an (X, Y )-path-system P with P ⊥ S. We denote the set of Erdős-Menger (X, Y )-path-systems by P D (X, Y ) and the set of Erdős-Menger (X, Y )-separations by S D (X, Y ).
Often, for a rooted D and v ∈ V (D) − r, we will want to consider Erdős-Menger (r, v)path-systems and -separations. If rv is an edge, then these are not defined above, and instead we will want to consider such path-systems and separations in D − rv. Hence, both to simplify and unify notation, we will use P D (v) and S D (v) as an abbreviation We define a partial order on S D (X, Y ) as follows: We write S T if S is an (X, T )separation.
We write G D (v) for the set of those subsets I of the ingoing edges of v for which there is an internally disjoint path-system from r with last edges I. More formally: In terms of this notation, we can define a flame as a rooted digraph F in which We will need an adaption of the following theorem due to Pym.

Theorem 2.2 (Pym). [8]
Let D be a digraph, let X, Y ⊆ V (D), and let P and Q be (X, Y )-

path-systems in D. Then there is an
Proof. After subdividing the ingoing edges of y, calling the set of new vertices Y , we can apply Theorem 2.2 to obtain the (X, Y )-path-system, which can be similarly translated back to an (X, y)-path-system with the desired properties.
The following Lemma is a corollary of Pym's Theorem, and shows that if I ∈ G D−rv (v), then the witness for it can be chosen from P D (v).

Lemma 2.4. Let D be a rooted digraph and let
Proof. By the definition of S D (v) there is a P ∈ P D (v) orthogonal to S. Let Q be an (r, v)path-system witnessing I ∈ G D (v). Let P ′ and Q ′ consist of the terminal segments of the paths P and Q from the last common vertex with S respectively. Applying Corollary 2.3 The concatenation of the initial segments of the paths P until S with the paths R ′ is a desired R.
Lastly, we will use the standard tool of the augmenting walk method.
For more details about the augmenting walk lemma and its role in the proof of the

Let D be a rooted digraph and let
Note that a D-large subgraph in particular contains out D (r). When the ambient digraph is clear from the context we will simply call L large.
We obtain the following corollary from Lemma 2.4.
Proof. By the definition of v-largeness, there exists some S ∈ S D (v) ∩ S L (v). We apply Lemma 2.4 with L, v, I − rv and S and extend the resulting path-system with the trivial path {rv} if rv ∈ I.
We will also need the following results about largeness from [5]. The first one gives us a condition for when largeness with respect to a subgraph G of D implies largeness with respect to D.
The second one says that to verify that L is large it is sufficient to show v-largeness only for vertices whose in-neighbourhoods differ in L and D.

Quasi-flames.
We extend the notion of a quasi-flame as defined in [5]. Let D be a rooted digraph, has the G-quasi-flame property at each vertex. For example the digraph in Figure 1 is a G-quasi-flame for every spanning subgraph G with countably many edges.
Our aim is to reduce the main theorem to the case where D is an F -quasi-flame.
Proof. Observe that F itself is an F -quasi-flame and the union of an ⊆-increasing nonempty chain of F -quasi-flames in D is an F -quasi-flame. Thus by Zorn's Lemma we may To show that Z has the desired properties, we apply Lemma 2.7. Assume that uv ∈ E(D) \ E(Z). By the maximality of Z, Z + uv is not an F -quasi-flame. The only possible reason for this is the existence of an For a countable G-quasi-flame D and a vertex v ∈ V (D), the following key lemma allows us to find a set I * ∈ G D (v) such that the restriction of D to I * at v is still a G-quasi-flame.
Then there is an We will prove this key lemma in Section 5 after laying the necessary ground work in Section 4.

Proof of the main theorem
In this section we prove our main result Theorem 1.4. To do so we assume Lemma 2.10, whose proof is postponed to Section 5.
Proof. Note that the union of any flame F in a given rooted digraph D with out D (r) is still a flame. Hence we may assume that F contains out D (r). By Corollary 2.9 we can also assume without loss of generality that D is an F -quasi-flame.
Let {v n : n ∈ N} be an enumeration of V (D) − r. Let G −1 := F and L −1 := D. We will recursively construct • a sequence (P n : n ∈ N) of Erdős-Menger path-systems P n ∈ P D (v n ); • two sequences (G n : n ∈ N) and (L n : n ∈ N) of spanning subdigraphs of D; such that for all n ∈ N: (2) G n is obtained by adding the edges of the paths in P n to G n−1 ; (3) L n = L n−1 ↾ vn in Gn (v n ); (4) L n is a G n -quasi-flame.
By (1), (2) and (3), (G n : n ∈ N) is ⊆-increasing, (L n : n ∈ N) is ⊆-decreasing and G n ⊆ L n for all n ∈ N. Moreover, Hence n∈N G n = n∈N L n =: F * , and F ⊆ F * . Furthermore, P n ∈ P D (v n ) ensures that F * is v n -large and in F * (v n ) ∈ G F * (v n ). Combining these we conclude that F * is a large flame in D extending F , as desired.
Suppose that G n−1 and L n−1 are defined for some n ∈ N and P i is defined for all i with 0 ≤ i < n. By Lemma 2.10 applied to L n−1 , G n−1 and v n there is an I * ∈ G L n−1 (v n ) such that L n−1 ↾ vn I * is a G n−1 -quasi-flame.
Since P i ∈ P L n−1 (v i ) for all i < n, properties (2) and (3) imply that L n−1 satisfies the conditions of Lemma 2.8 and hence L n−1 is large. Thus, we may apply Corollary 2.6 to L n−1 , v n and I * ∈ G L n−1 (v n ) to find an Erdős-Menger path-system P n ∈ P D (v n ) ∩ P L n−1 (v n ) such that I * ⊆ E + (P n ) ∪ {rv n }. We define G n and L n according to (2) and (3), i.e., G n := G n−1 ∪ P n and L n := L n−1 ↾ vn in Gn (v n ). We claim that G n , L n and P n satisfy properties (1)-(4).
Clearly, properties (2) and (3) are satisfied. By construction, P n ⊆ L n−1 and Finally, we need to show that L n is a G n -quasi-flame.
Note that by construction in Ln (v n ) ∈ G Ln (v n ), as guaranteed by P n . Hence we only need to check that the G n -quasi-flame property holds at v m = v n . However, L n−1 ↾ vn I * is a G n−1 -quasi-flame and hence, since I * ⊆ in Gn (v n ), it follows that L n satisfies the G n−1quasi-flame property at every v m = v n . Furthermore, since in Gn (v m ) has at most one new edge compared to in G n−1 (v m ), it follows that L n satisfies the G n -quasi-flame property at

Incompressibility
Incompressibility will be a key concept in the proof of Lemma 2.10 but its scope of potential applications is not restricted to flames. For example it played an important role in the arborescence packing result [4] of the third author.
We will be interested in when we can find an (X, Y )-path-system which covers the vertices of X, in which case we say that X is joinable to Y . If X and Y are both infinite, then clearly this cannot happen if there is some finite (X, Y )-separation in D. However, even when the two sets are infinitely connected, it can happen that X is not joinable to Y .
One obstruction to this would be an (X, Y )-separation S which is 'smaller than X' in the sense that it is the 'same size' as a proper subset of X. For finite sets U and W it is clear that the equality |U| = |W | is equivalent to the condition that every injective map from U to W is also a bijection. For countably infinite sets this is no longer true, since U can be injected, or 'compressed', into an arbitrary infinite subset of W . However if we place restrictions on the types of injections we allow, then there may be sets U and W where only bijections are possible, and so U is 'incompressibile' to W , respresenting a sort of tightness between U and W with respect to the set of injections.
In our particular case we are considering the injections which arise from (U, W )-pathsystems which cover the vertices in U. Each such system defines in a natural way an injection from U to W , and for certain pairs of infinite sets, the only injections which arise in this manner will be bijections. In this case we will say that U is incompressible to W .
Then, an (X, Y )-separation S can be the 'same size' as a proper subset X ′ of X, if X ′ is incompressible to S. It is then easy to see this is an obstruction to X being joinable to Y , as any (X, Y )-path-system witnessing this would contain an (X ′ , S)-path-system, which by the incompressability of X ′ to S would cover S, and so separate X from Y .
We will show that, if there is some ⊆-maximal proper subset X ′ of X which is joinable to Y , then this will in fact by witnessed by such a 'tight' (X, Y )-separation S, that is, where X ′ is incompressible to S.
We denote the set of all such path-systems by J D (X, Y ).
For examples of these terms, see Figure 4.  in each example is the set of sources, the set Y the set of sinks. In (a), X is not joinable to Y . In (b) and (c), X is joinable to Y but not incompressible.
In fact in (b) no (X, Y )-path system covers Y , while in (c) there exists a perfect matching. In (d), X is incompressible to Y . Note that in (a), every subset X ′ of X for which X \ X ′ is infinite is finitely extenable.
Let D, X and Y be fixed as above. For technical reasons we assume that the vertices in X have no ingoing edges and the vertices in Y have no outgoing edges. Those edges will never be contained in any (X, Y )-path and hence to not play any role in any of the statements.

Lemma 4.2. Let
x ∈ X and assume that (X − x) is joinable to Y but X is not. Then Proof. The Aharoni-Berger theorem [1] ensures that S D (X, Y ) = ∅. Let S be thesmallest element of S D (X, Y ), which exists by Lemma 2.1, and let P ∈ J(X − x, S) be arbitrary. Since there is a P ′ ∈ P D (X, Y ) orthogonal to S, if X is joinable to S then we can extend an (X, S)-path-system using the terminal segments of P ′ starting at S to one witnessing that X is joinable to Y . Hence, X is not joinable to S.
The vertices in X have no ingoing edges by assumption, and hence none of the paths in P goes through x. Applying Lemma 2.5 to X, S and P we see that the first case of the lemma would allow us to conclude that X is joinable to Y and hence there is some (X, S)-separation T such that P ⊥ T . Since T is also an (X, Y )-separation, it follows that T ∈ S D (X, Y ) and hence by -minimality of S, we have S = T and so V + (P) = S.
Proof. It is enough to prove the case where |X \ X ′ | = 1, since the general case then follows by applying this recursively.
Let P ∈ J(X ′ , Y ′ ). We apply Lemma 2.5 to X, Y and P. If the first case of the lemma holds, then we augment to an (X, Y )-path-system P ′ which witnesses that X is joinable Otherwise, there is some (X, Y )-separation S with S ⊥ P. However, since X is joinable to Y and S is an (X, Y )-separation, it follows that X is joinable to S. But then by extending an (X, S)-path-system witnessing that X is joinable to S with the terminal segments of P from S to Y ′ we get an (X, Y )-path-system witnessing that X is joinable to Y ′ .

Lemma 4.4. Let D ′ be a subgraph of D obtained by deleting finitely many vertices of D
and then deleting finitely many edges from the resulting digraph. If X is incompressible to Y in D ′ , then X was already incompressible to Y in D.
Proof. Suppose for a contradiction that there is some P ∈ J D (X, Y ) for which Y * := V + (P) is a proper subset of Y . Since the paths in P are vertex disjoint, all but finitely many of the paths in P are contained in D ′ . Let P ′ = {P ∈ P : P ⊆ D ′ }. Then there is some Y ′ ⊆ Y * and X ′ ⊆ X such that P ′ ∈ J D ′ (X ′ , Y ′ ) and However, then Proof. Assume, for a contradiction, that there is some P ∈ J(X, Y ) with Y \ V + (P) = ∅.
Let P ∈ P be the path starting in x. Since X − x is incompressible to Y − y there is some Q ∈ P with P = Q which ends in y. Let P ′ = P \ {P, Q}.
Definition 4.6. Let D be a digraph and let X, A natural question is whether this implies that O is joinable to Y for some X ′ ⊆ O ⊆ X with |O \ X ′ | = ℵ 0 . We will actually need a stronger statement, given a countable collection of infinite subsets of X \ X ′ we want to find such an O which meets all of them.
To do so we will need to establish the following lemma first.
Proof. It is enough to prove the special case where U = {v} for some v ∈ V (D), the general case follows by applying it to the vertices in U one by one recursively.
If v ∈ X, then W := {v} is suitable. Indeed, since the vertices in X have no ingoing edges, the deletion of v cannot ruin anything in this case. So, let us suppose that v / ∈ X. We may assume that there is a set {x i : 0 ≤ i ≤ ℓ} ⊆ X \ X ′ for which Without loss of generality we can assume that ℓ = 0 since otherwise we replace X ′ with X ′ ∪ {x i : 0 ≤ i < ℓ}. In the light of this we write simply x instead of x 0 and show that W := {v, x} satisfies the conclusion of the lemma.

By Lemma 4.2 there is an
However, then by Lemma 4.4, X ′ is incompressible to S in D as well. Note that S + v separates Y from X ′ in D and by assumption X ′ + x is joinable to Y in D.
Hence, X ′ + x is joinable to S + v in D. Then, by Lemma 4.5 applied to x ∈ X ′ + x and v ∈ S + v we can conclude that X ′ + x is also incompressible to S + v in D.
Let an arbitrary such O be fixed (see Figure 4.2). Since X ′ is However, since X ′ + x is incompressible to S + v, and S + v separates Y from X ′ + x, the subset of paths starting at vertices of X ′ + x In particular, no path in P \ P ′ meets S + v, and therefore none of these paths meets any vertex from which Y is separated by S + v in D.
Let Q ∈ J D−v (X ′ , S). By the above comment, no path in Q meets any path in P \ P ′ and so we can form a set of (X ′ , Y )-paths disjoint from the paths in P \ P ′ by extending each path in Q by the terminal segment of a path in P ′ after S.

This set of paths together with
Using this we can show the following lemma.

Lemma 4.8. Assume that a countable
Proof. We build the desired (X, Y )-path-system by recursion. The key to accomplish each step is the following claim.

Claim 1.
For every x ∈ X there exists an (x, Y )-path P for which there is a finite vertex set W ⊇ V (P ) with W ∩ X ′ ⊆ {x} such that the deletion of W preserves the conditions of Lemma 4.8 for the remaining system, i.e., ( Proof. Since X ′ ⊆ X is (X, Y )-finitely extendable in D we can pick a P ∈ J(X ′ + x, Y ). Let P be the unique element of P with first vertex x. We obtain W by applying Lemma 4.7 with U = V (P ) and X ′ if x / ∈ X ′ . If x ∈ X ′ , then we apply Lemma 4.7 with V (P ) and X ′ − x.
First we assume that |X ′ | = ℵ 0 , and we enumerate X ′ = {x k : k ∈ N}. We will build an (X, Y )-path-system P = (P i : i ∈ N) where P 2k starts at x k and P 2k+1 starts in V k . We also will maintain a system D n , X n , Y n obtained from D, X, Y by the deletion of a finite vertex set containing i<n V (P i ). We will also demand that if n ≤ 2k, then x k ∈ X n , and, moreover, that X ′ ∩ X n is (X n , Y n )-finitely extendable in D n . Then O := V − (P) will satisfy the Lemma 4.8.
At the beginning we take D 0 := D, X 0 := X, Y 0 := Y and we do not have any paths defined. In step n we apply Claim 1 with D n , X n , Y n and X ′ ∩ X n choosing x to be x 2k if n = 2k and an arbitrary element of V k ∩ X n if n = 2k + 1. This yields the desired path P n . The triple D n+1 , X n+1 , Y n+1 is obtained by the deletion of V (P n ) together with the finitely many extra vertices given by Claim 1.
For a finite X ′ we proceed similarly except only |X ′ | many steps are devoted to join X ′ to Y .

Applications of incompressibility.
For this subsection, let D be a digraph rooted in r. Suppose P I witnesses I ∈ G D (w). The edge uv must be in one of the paths, say in P i 0 ∈ P I where i 0 ∈ I is the last edge of P i 0 . Let A, X, Y and X I as in the proof of Corollary 4.9.
Note that X I − x i 0 is joinable to Y in A − vu and X I is joinable to Y in A but not in A − vu.
It follows from Lemma 4.2 that there is some S ′ ∈ S A−vu (X I , Y ) such that X I − x i 0 is incompressible to S ′ in A − vu. We may assume that S ′ ∩ (X I ∪ Y ) = ∅, since otherwise we replace in S ′ each x ∈ S ′ ∩ X I by its unique out-neighbour in A and each y ∈ S ′ ∩ Y by its unique in-neighbour in A. Now X I − x i 0 is also incompressible to S ′ in A by Lemma 4.4, but S ′ does not separate Y from X I in A since otherwise X I would not be joinable By translating back the results to the original digraph we have the following conclusions.
• Every path-system witnessing I ∈ G D (w) must be orthogonal to S ′′ .
• For any in-neighbour u ′ = u of v, any (r, u ′ )-path Q in D which avoids S ′′ (should one exist) must necessarily contain w.
If S ′′ separates the tails of in D (v) − u from r and Q is a path-system witnessing I ∈ G D (w), then the set P of initial segments of the paths in Q until S ′′ satisfies the conclusion of the lemma with S := S ′′ .
Otherwise, let f ∈ in D (w) \ I be arbitrary. By the second condition S ′′ + w separates the tails of in D (v) − u from r (see Figure 4.1) and by assumption there is some pathsystem R witnessing that (I + f ) ∈ G D (w). Since the set of paths in R with last edge in I is orthogonal to S ′′ , the set P of initial segments of the paths in R until S ′′ + w is as The separator S ′′ + w and path-system P.

Proof of Lemma 2.10
We will use our results about incompressibility from Subsection 4.1 to prove Lemma 2.10. Then there is an Proof. We may assume without the loss of generality that out D (r) ⊆ E(G) since otherwise we may add these edges to G while D remains a G-quasi-flame for the new G.
Firstly, let us analyse how D ↾ v J can fail to be a G-quasi-flame for some with |I \ in G (w)| < ℵ 0 and I / ∈ G D 0 (w). Note that there are only countably many relevant sets. Any path-system witnessing I ∈ G D (w) for some relevant I necessarily uses at most one edge from in D (v) \ in G (v). Therefore for every relevant I the set is non-empty. Moreover, for a J ∈ G D (v) with J ⊇ in G (v) the following statements are equivalent: • D ↾ v J is an a G-quasi-flame; • N I ∩ J = ∅ for every relevant I.
By applying Corollary 4.9 with D, in G (v) ∈ G D (v) and {N I : |N I | = ∞}, we obtain first Let I * be a superset of I * 0 that intersects all the finite N I and is ⊆-minimal with respect to this property. Clearly, the definition of I * ensures the G-quasi-flame property of D * := D ↾ v I * at all the vertices other than v. It remains to show that I * ∈ G D * (v).
We assume I * I * 0 , since otherwise there is nothing to prove, and fix an enumera- Recall that D * has the G-quasi-flame property at w. Moreover, u i = r since other- would contradict e i ∈ I * \ I * 0 . Therefore we can apply Lemma 4.10 with D * , I i ∈ G D * (w) and the edge e i , and we denote by S ′ i and P i the resulting set and (r, S ′ i )-path-system, respectively. By using the path-systems P i for e i ∈ I * \ I * 0 we will build sets Q and R of paths in D * with the following properties. (ii) Q is an (r, v)-path-system with E + (Q) = I * \ I * 0 ; (iii) R is an (r, S − v)-path-system; (iv) V (R) ∩ V (Q) = {r} for all Q ∈ Q and all R ∈ R.
First we show that I * ∈ G D * (v) if we have such sets Q and R. Let P be a path-system witnessing I * 0 ∈ G D * (v). Then each P ∈ P has a last common vertex v P with S − v by (i). We extend the unique R ∈ R that terminates at v P with the terminal segment of P from v P to v. The set of these extended paths united with Q witness I * ∈ G D * (v). So let us build the path-systems Q and R. We construct two sequences (Q k : 0 ≤ k < |I * \ I * 0 |) and (R k : 0 ≤ k < |I * \ I * 0 |) of sets of paths such that: (1) S k := V + (Q k ∪ R k ) separates the tails of the edges in D * (v) \ {e i : 0 ≤ i ≤ k} from r in D * where r / ∈ S k ; (2) Q k is an (r, v)-path-system with E + (Q k ) = {e i : 0 ≤ i ≤ k}; (3) R k is an (r, S k − v)-path-system; (4) V (R) ∩ V (Q) = {r} for all Q ∈ Q k and all R ∈ R k ; (5) Q k ⊆ Q k+1 ; and (6) each path in R k+1 extends some path in R k (not necessarily properly). For k = 0, let Q 0 be the unique path in P 0 with E + (Q 0 ) = {e 0 } (see Lemma 4.10).
Then Q 0 := {Q 0 } and R 0 := P 0 − Q 0 satisfy the conditions. Let us assume that we have constructed Q k and R k . Recall that the tail of e k+1 is separated from r by S k because of (1). Thus, each P ∈ P k+1 whose terminal vertex is separated from r by S k meets S k − v. Let us denote the last common vertex of such a P with S k − v by v P . We take the unique R ∈ R k terminating v P and extend it forward by the terminal segment of P from v P . The unique newly constructed path with last edge e k+1 together with Q k define the set Q k+1 , and the other (not necessarily properly) forward-extended paths in R k define R k+1 .
For all the conditions but (1) it follows from the construction directly that Q k+1 and R k+1 satisfy them. To show the preservation of (1) we need the following lemma. Proof. Suppose for a contradiction that some (r, x)-path P avoids S. Let S 0 ∈ S be arbitrary and let s 0 be the last common vertex of P with S 0 . We conclude that s 0 / ∈ S.
Hence, there is some S 1 ∈ S that does not separate s 0 from r. Then the last common vertex s 1 of P with S 1 is strictly later on P than s 0 . By continuing the construction recursively we end up with infinitely many pairwise distinct s n ∈ V (P ) which is a contradiction.
Both S ′ k+1 and S k separate the tails of the edges in the set in D * (v) \ {e i : 0 ≤ i ≤ k + 1} from r in D * . Moreover, s ∈ S ′ k+1 \ S k+1 only if s is not separated from r by S k , and hence Therefore (1) follows by applying Lemma 5.1 to the tails of the edges e i for all i > k + 1 separately.
If |I * \ I * 0 | = k + 1 ∈ N, then Q k and R k are the desired Q and R. Suppose that |I * \ I * 0 | = ℵ 0 . Then let Q := k∈N Q k and R := m∈N m≤k∈N R k . For all the conditions but (i) it follows from the construction directly that Q and R satisfy them. By Lemma 5.1, {S k : k ∈ N} separates the tails of the edges in the set I * 0 from r in D * . Note that if an R ∈ R k is properly extended to a path in R k+1 in step k then its last vertex is not separated by S k+1 from r. Thus {S k : k ∈ N} ⊆ V + (R) from which (i) follows.