Egerváry Research Group
on Combinatorial Optimization
Technical reportS
TR-2014-13. Published by the Egerváry Research Group, Pázmány P. sétány 1/C,
H1117, Budapest, Hungary. Web site:
www.cs.elte.hu/egres .
ISSN 15874451.
Edmonds' Branching Theorem in Digraphs
without Forward-innite Paths
Attila Joó
2014
EGRES Technical Report No.
2014-13
1
Edmonds' Branching Theorem in Digraphs without
Forward-innite Paths
Attila Joó
Abstract
By a well-known theorem of Edmonds if a nite digraph is k-edge-connected
from vertex r then it has k edge-disjoint spanning arborescences rooted at r.
As it was shown by R. Aharoni and C. Thomassen in [7] this does not remain
true for innite digraphs. Thomassen also proved that for the class of digraphs
without backward-innite paths the above theorem of Edmonds remains true.
Our main result is that for digraphs without forward-innite paths the theorem
is also true, even in general form.
1
Notions
Digraphs
D = (V, A)
considered here may have multiple edges and arbitrary size.
B ⊆ V then we write D[B]
B . For X ⊆ V let inD (X) and outD (X) be the
of X in D , and %D (X), δD (X) their cardinalities,
Loops are also allowed but irrelevant to our subject. If
for the subgraph of
D
spanned by
sets of ingoing and outgoing edges
respectively.
By a path we mean a directed, possibly innite, simple path so that
start(P ) and end(P ) denotes the rst and last
point of the path P if it exists. For e = (x, y) let start(e) = x and end(e) = y . For
X, Y ⊆ V let eD (X, Y ) = {e ∈ A : start(e) ∈ X, end(e) ∈ Y }, for singletons we write
e(x, y) instead of e({x}, {y}). We say that the path P goes from X to Y if V (P )∩X =
{start(P )} and V (P ) ∩ Y = {end(P )} (start(P ) = end(P ) is allowed). For singletons
we write here also v instead of {v}. Call a digraph A = (V, A) an arborescence with
root vertex r if A is a directed tree such that all vertices are reachable from r . We
call min{%D (X) : ∅ 6= X ⊆ V \ {r}} the edge-connectivity of D from r and we say
D is κ-edge-connected from r if this cardinal is at least κ. For an undirected graph
G denote by V (G) and E(G) its point-set and edge-set respectively. If G is directed
we write A(G) instead of E(G). If A is a subarborescence of D and e ∈ outD (V (A))
then sometimes we write A + e instead of (V (A) ∪ {end(e)}, A(A) ∪ {e}).
repetition of vertices is not allowed.
2
Introduction
Edmonds proved in [1] his famous branching theorem which states that if a nite
digraph is
k -edge-connected from r
then it has
k
edge-disjoint spanning arborescences
2014
2
Section 3. Main result
rooted at
r.
Lovász gave a new elegant proof for this theorem of Edmonds in [2] and
his techniques opened the door for further generalizations such as [3], [4], [5] and [6].
Innite generalizations have been marred by a negative result by R. Aharoni and C.
Thomassen [7]. They constructed, for any
k ∈ N,
a countably-innite, locally nite,
G such that G has k -connected orientations and G has vertices u, v such
that deleting the edges of an arbitrary u − v path disconnects G.
Thomassen showed (unpublished) that if D = (V, A) does not contain backwardinnite paths and it is k -edge-connected from r then it has k edge-disjoint spanning
arborescences rooted at r . The main idea of his proof is the following. First one
0
0
0
constructs a spanning subgraph D = (V, A ) such that D is also k -edge-connected
0
from r and all points of D have nite in-degrees. After that one constructs the
0
desired arborescences in D using the nite version of the theorem and compactness
simple graph
arguments.
Our main result is that disallowance of the forward-innite paths instead of backwardinnite paths is also sucient. The proof is longer and uses totally dierent techniques
than the sketch of Thomassen's proof above.
3
Main result
In this section we state and prove our main result theorem 3.1.
Let D = (V, A) be a digraph, κ a (nite or innite) cardinal and
Ai = (Vi , Ai ) (i < κ) edge-disjoint subarborescences of D such that for all v ∈ V the
set Hv = {i < κ : v ∈
/ Vi } is nite and let Df = (V, A \ ∪i<κ Ai ). Suppose that Df
does not contain forward-innite paths. Then these subarborescences can be extended
to edge-disjoint spanning arborescences of D without changing their roots if and only
if
Theorem 3.1.
∀X (∅ 6= X ⊆ V =⇒ %Df (X) ≥ |{i : Vi ∩ X = ∅}|).
(1)
Let us formulate two important special case of the theorem above.
Corollary 3.2. Let D = (V, A) be a digraph, k ∈ N and Ai = (Vi , Ai ) (i < k)
edge-disjoint subarborescences of D and let Df = (V, A \ ∪i<k Ai ). Suppose Df does
not contain forward-innite paths. Then these subarborescences can be extended to
edge-disjoint spanning arborescences of D without changing their roots if and only if
condition (1) holds.
Let the digraph D be k -edge-connected from the vertex r for some
k ∈ N and suppose that there are no forward-innite paths in D. Then there are k
edge-disjoint spanning arborescences in D rooted at r.
Corollary 3.3.
Remark 3.4. Theorem 3.1 remains true if we change forward-innite to backward-
innite because Thomassen's proof (see the sketch in the introduction) works also in
this general situation.
EGRES Technical Report No.
2014-13
3
Section 3. Main result
Proof of Theorem 3.1.
{Ai }i<κ
Without loss of generality we can assume that the roots of the
arborescences are the same point by adding
vertex and
ri
is the original root of
Ai
and extend
rri
Ai
D, where r is a new
rri (i < κ). The necessity
edges to
with
of the condition is obvious, so we show only that it is sucient. To do so, we need
the following lemma.
For any ζ < κ and v ∈
/ Vζ there is a path P in Df from Vζ to v
0
0
such
( that condition (1) holds for D and the arborescence-system {Ai }i<κ where Ai =
Ai + P if i = ζ
.
Ai
otherwise
Lemma 3.5.
Before the proof of the lemma 3.5 we need to build up some basic tools in spirit of
Lovász's proof for the nite version of the theorem in [2].
Call a set
∅ 6= X ⊆ V accurate if %Df (X) = |{i : Vi ∩ X = ∅}| and dangerous
X ∩ V0 6= ∅. It is easy to see that if e ∈ outDf (V0 ) then the
if it is accurate and
extension
def
A0 = A0 + e
violates condition (1) if and only if
e
is an ingoing edge of
some dangerous set.
Proposition 3.6.
If X, Y are dangerous and X ∩ Y 6= ∅ then X ∩ Y is dangerous.
Proof:
Let s : P(V ) → N, s(X) = |{i : Vi ∩ X = ∅}|. Then s is supermodular i.e.
X, Y ⊆ V we have s(X) + s(Y ) ≤ s(X ∪ Y ) + s(X ∩ Y ). Indeed, let i < κ be
arbitrary. If Vi ∩ X = ∅ and Vi ∩ Y = ∅ then Vi ∩ (X ∪ Y ) = ∅ and Vi ∩ X ∩ Y = ∅, so
the Vi 's contribution to both sides of the inequality is 2. If Vi ∩ X = ∅ and Vi ∩ Y 6= ∅
then the Vi 's contribution to both sides is 1.
Observe that equality holds if and only if there does not exist any Vi such that Vi ∩X 6=
∅, Vi ∩ Y 6= ∅ and Vi ∩ X ∩ Y = ∅. Let p(X) = %Df (X) − s(X). Then condition (1) is
equivalent with p(X) ≥ 0 for all X 6= ∅, and accurateness of X means p(X) = 0. The
function %Df is submodular, so p is too i.e. p(X) + p(Y ) ≥ p(X ∪ Y ) + p(X ∩ Y ) holds
for all X, Y ⊆ V . Let X, Y be dangerous and X ∩ Y 6= ∅. Then by submodularity
for
and condition (1), we get
0 + 0 = p(X) + p(Y ) ≥ p(X ∪ Y ) + p(X ∩ Y ) ≥ 0 + 0,
so
X ∪Y
and
X ∩Y
the observation about the function
X ∩ Y ∩ V0 6= ∅,
so
s(X) + s(Y ) = s(X ∪ Y ) + s(X ∩ Y ). By
by X ∩ V0 6= ∅, Y ∩ V0 6= ∅ it follows that
are accurate therefore
X ∩Y
s
and
is dangerous.
Let B be a dangerous set. Then for any w ∈ B there is a path R
from V0 ∩ B to w in Df [B].
Proposition 3.7.
Proof:
B0
be the set of vertices which are reachable from V0 ∩ B in Df [B].
0
0
Suppose, seeking a contradiction, that B 6= B . Then B \ B violates condition (1)
Let
which is a contradiction.
Proposition 3.8. For all w ∈ V there is a system of edge-disjoint paths {Pi }i∈H in
w
Df such that Pi goes from Vi to w.
EGRES Technical Report No.
2014-13
4
Section 3. Main result
Proof:
Df to Df0 by adding new vertices and edges (see gure 1). Let
V (Df0 ) = V ∪ {s} ∪ {vi }i∈Hw , |eDf0 (s, vi )| = 1 (i ∈ Hw ) and |eDf0 (vi , u)| = ℵ0 (i ∈
Hw , u ∈ Vi ). If there are |Hw |-many edge-disjoint paths from s to w in Df0 then we
We extend
are done.
Suppose, seeking contradiction, that there aren't. By Menger's theorem
w ∈ X ⊆ V (Df0 ) \ {s} with %Df0 (X) < |Hw |. Let l = {vi }i∈Hw \ X . Note
that 0 < l otherwise svi ∈ inD 0 (X) (i ∈ Hw ) and hence |Hw | ≤ %D 0 (X) would follow.
f
f
By the innitely many parallel edges X ∩ V is disjoint from at least l arborescences.
Otherwise %D 0 (X) = %Df (X ∩ V ) + |Hw | − l so %Df (X ∩ V ) = l + %D 0 (X) − |Hw | < l
f
f
but then X ∩ V violates condition (1) in D which is a contradiction.
there is a
V1
X
V0
s
v1
v0
v2
Figure 1: The construction of
Hw = {0, 1, 2}, l = 1.
Df0
V2
and the cut
X
from the proof above in the case
(Thick arrows stand for countably innite parallel edges).
Let B2 ⊆ B1 be dangerous sets. Denote the endpoints of edges
in
in
(B
)
with
multiplicity
by s1 , . . . , sm (where the multiplicity of a vertex z is
Df
1
inD (B1 ) ∩ inD (z)). Then there is a system of edge-disjoint paths {Pj }m in Df [B1 ]
f
f
j=1
such that Pj goes from sj to B2 . If {Pj }m
is
a
path-system
like
above
and
%Df (B2 ) =
j=1
m
%Df (B1 ) then eDf (B1 \ B2 , B2 ) ⊆ ∪j=1 A(Pi ).
Proposition 3.9.
Proof:
Extend
Df [B1 ]
with a new vertex
s
part of the proposition it is enough to show that the
connected from
s.
ss1 , . . . , ssm . For the rst
resulting graph H is m-edge-
and new edges
Suppose, seeking contradiction, that it is not. Then by Menger's
∅ 6= X ⊆ B1 such that %H (X) < m. Using dangerousness of B1
m = s(B1 ) ≤ s(X) so %Df (X) = %H (X) < m ≤ s(X) thus X violates
theorem there is a
we obtain
condition (1) which is a contradiction.
def
m
Let {Pj }j=1 be a desired path system and l = |{j : sj
exactly l of the pahts above that have 0 length, the other
edges of
eDf (B1 \ B2 , B2 ).
Otherwise
∈ B2 }|. Then there are
m − l paths use one-one
eDf (B1 \ B2 , B2 ) = %Df (B2 ) − l = m − l.
Corollary 3.10. Let B be a dangerous set and e ∈ inDf (B). Then for any w ∈ B
there is a path R from end(e) to w in Df [B].
EGRES Technical Report No.
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Section 3. Main result
Proof of Lemma 3.5.
By symmetry it is sucient to prove the lemma for
ζ = 0.
Let
us introduce the following recursive process (informal rst). Here we use the notions
A0 , Df
as variables.
P1 from V0
P1 and their
Fix a path
the edges of
to
v
in
Df
(it exists by proposition 3.7). Extend
A0
with
endpoints one by one in sequence until it is doable without
e1 is the rst edge that we can not add to A0 then
B1 such that e1 ∈ inDf (B1 ). Fix a path P2 that goes from
Df [B1 ] (it exists by proposition 3.7). Extend A0 with the edges
the violation of condition (1). If
there is a dangerous set
V0 ∩ B1 to end(e1 ) in
of P2 and their endpoints
one by one in a sequence until it does not violate condition
(1). If they never violate condition (1) we reach the vertex
and then we step
P1 which is
e2 ∈ A(P2 ) be the rst violating edge. Then e2 ∈ inDf (B2 ) for
some dangerous set B2 and we can assume, by proposition 3.6, that B2 ⊆ B1 . We
take a P3 path from V0 ∩ B2 to end(e2 ) in Df [B2 ] and continue. . .
Surprisingly, with this naive process we reach the vertex v after nitely many steps.
back to path
in
V0 .
P1
end(e1 ),
and continue the process starting with the last vertex of
Otherwise let
To show this we make precise the informal description above. We dene by recursion
n
n
n
n
a sequence (A0 , Tn , xn ) where A0 = (V0 , A0 ) is the extension of the arborescence
A0 after the n-th step, Tn is a rooted tree with root vertex s and xn ∈ V (Tn ).
def
Ani = (Vin , Ani ) = Ai
def
n
i<κ Ai ). Call B ⊆ V nn
dangerous if it is dangerous with respect to the arborescence-system {Ai }i<κ . Let
S∞
S∞
T = ( n=0 V (Tn ), n=0 E(Tn )). Denote by b(t) the smallest n such that t ∈ V (Tn ).
The vertices of the trees Tn are in the form t = (Bt , Pt , et ) (except the root which is
b(t)
s = (Bs , Ps )) where Bt ⊆ V is an b(t)-dangerous set, Pt is path from V0 to end(et ) in
b(t)
Df [Bt ] and et ∈ inDb(t) (Bt ). Denote by parentT (t), childT (t) the parent of the vertex
Let
if
0 < i
and
Dfn = (V, A \
S
f
t
and the set of children of vertex t in the tree T respectively.
0
In the beginning let A0 = A0 , T0 has only its root s = (Bs , Ps ) where Bs = V and
Ps is an arbitrary path from V0 to v in Df (it exists by proposition 3.7) and x0 = s.
The recursion is as follows
v ∈ V0n
Case 1.
def
def
def
An+1
= An0 , Tn+1 = Tn , xn+1 = xn
0
v∈
/ V0n
Case 2.
and
end(Pxn ) ∈ V0n
def
def
def
xn+1 = parentTn (xn ), An+1
= An0 , Tn+1 = Tn
0
v, end(Pxn ) ∈
/ V0n and the extension An0 + e does not violate condition
e ∈ A(Pxn ) is the outgoing edge of the last vertex of Pxn which is in V0n ,
Case 3.
where
def
def
(1)
def
An+1
= An0 + e, Tn+1 = Tn , xn+1 = xn
0
v, end(Pxn ) ∈
/ V0n and the extension An0 + e violates condition
e ∈ A(Pxn ) is the outgoing edge of the last vertex of Pxn which is in V0n .
Case 4.
Then
e ∈ inDfn (B)
a new vertex
t
for some
such that
t
n-dangerous B .
is a child of
xn .
Let
Let
Tn+1
be the extension of
def
Bt = B ∩ Bxn
EGRES Technical Report No.
(1) where
2014-13
(it is
Tn
with
n-dangerous
by
6
Section 3. Main result
Pt
proposition 3.6),
3.7),
def
et = e
and
a path from
Bt ∩ V0n
to
end(e)
Dfn [Bt ]
in
def
def
An+1
= An0 , xn+1 = t.
0
It is routine to check the following properties of the
1. If
(it exists by proposition
t2 ∈ childT (t1 )
then
(An0 , Tn , xn )n∈N
Bt2 ⊂ Bt1 (t1 , t2 ∈ V (T )).
2.
et ∈ A(PparentT (t) ) ∩ e(BparentT (t) \ Bt , Bt ) (t ∈ V (T ) \ {s}).
3.
et ∈ A(Dfn ) (t ∈ V (T ) \ {s}, n ∈ N)
4. If
5.
xn 6= xn+1
xb(t) = t
then
and if
Proposition 3.11.
Proof:
sequence.
{xn , xn+1 } ∈ E(T ) (n ∈ N).
t 6= s
then
xb(t)−1 = parentT (t).
If t1 , t2 ∈ childT (t) and b(t1 ) < b(t2 ), then et1 precedes et2 in Pt .
b(t )−1
end(et1 ) ∈ V0 2
(see case 4). Let m = min{n ∈
N : b(t1 ) ≤ n, xn = t}. Then m ≤ b(t2 ) − 1 because xb(t2 )−1 = t (see property 5) and
b(t1 ) ≤ b(t2 ) − 1. By xb(t1 ) = t1 and property 4 at the m − 1-th step of the recursion
m−1
arise case 2 and xm−1 = t1 . So by the denition of case 2 end(et1 ) ∈ V0
and so
bt2 −1
.
end(et1 ) ∈ V0
It is enough to show that
Corollary 3.12.
For all t ∈ V (T ) |childT (t)| ≤ |A(Pt )|.
We need to show that case 1 occurs in the recursion. Suppose, seeking a contradiction,
that it does not i.e. v ∈
/ V0n for all n.
We show that T is an innite tree. Suppose, seeking a contradiction, that it is
m. In all case 3-type step we put an edge of Pt to the
t ∈ V (Tm ), hence after the m-th step there are at most
t∈V (Tm ) |A(Pt )| case 3-type steps and at most height(Tm ) case 2-type steps, so there
must be a case 4-type step after the m-th step, thus Tm 6= T which is a contradiction.
So T is a countably-innite tree with nite degrees (see corollary 3.12). Then by
König's lemma there is an innite path (tn )n∈N in T such that tn+1 ∈ childT (tn ) (n ∈
N). We may assume that t0 6= s.
Let's look at the monotone decreasing sequence (Btn )n∈N (see property 1).
not. Then
0-th
P
T = Tm
for some
arborescence for some
Proposition 3.13.
Proof:
supn∈N s(Btn ) < ∞.
It is enough to show that
def
T∞
n=0 Btn 6= ∅, because if u
is a nite set and s(Btn ) ≤ |Hu |
C =
∈ C , then by
(n ∈ N). We
sets {Btn }n∈N and
Hu = {i < κ : v ∈
/ Vi }
R by recursion such that R enters into all of the
does not enter into any of the sets {V \ Btn }n∈N . R can not be forward-innite by
assumption so if we succeed, then end(R) ∈ C and the proof is completed. Let R0
b(tn+1 )
be the length 1 path et0 . By using corollary 3.10 with the arborescences {Ai
}i<κ
b(tn+1 )
[Btn ] from end(Rn ) to Bn+1 . By connecting the paths
there is a path Rn+1 in Df
{Rn }n∈N we get the desired path R.
assumption
construct a path
EGRES Technical Report No.
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Section 3. Main result
s(Btn ) is bounded (see proposition 3.13). By deleting few initial elements of the sequence (tn )n∈N we may assume that the sequence
s(Btn ) is constant, i.e. the sets Btn are disjoint from the same l-many Vi 's. Let
sn (X) = |{i : Vin ∩ X = ∅}|. Then the sequence sb(tn ) (Btn ) is also constant, namely
l − 1 or l depending on whether the sets Btn are disjoint from V0 or not. Denote
this number by m. Note that m > 0 because etn ∈ in b(tn ) (Btn ) and Btn is b(tn )Df
dangerous. Apply proposition 3.9 to the sets Btn and Btn+1 with the arborescenceb(tn+1 )
system {Ai
}i<κ (n = 0, 1, . . . ). By connecting the results we get a system of
m
edge-disjoint paths {Pi }j=1 entering all of the sets {Btn }n∈N . The edges in {etn }n∈N
are pairwise distinct because start(etn+1 ) ∈ Btn \ Btn+1 (see property 2) and by the
Sm
second part of proposition 3.9 and by properties 2, 3 {etn }n∈N ⊆
j=1 A(Pi ). So at
The monotone increasing sequence
least one of the paths are innite which contradicts to our assumption.
Now we continue the proof of theorem 3.1. If
of the sets
Hv
v ∈ V then by lemma 3.5 and niteness
{Ai }i∈Hv without the violation of
we can extend the arborescences
condition (1) with nitely many new points and edges such that all of these extensions
contain
v.
In the countable case we can construct the desired spanning arborescenes
by an easy recursion. In the
n-th
step do the extensions above with the arborecences
∞
after the previous step and the following vertex vn where V = {vn }n=0 . In the
uncountable case we have to be more careful because we need to assure that we do
not violate condition (1) in limit steps. The main idea is that if we extend one of the
arborescences with some vertex
v,
then at the successor of these steps we put
all arboresences which missed it.
v
into
def
V = {vα }α<λ , where λ = |V |. We extend
α
α
α
the arborecences by transnite recursion on λ. Let Ai = (Vi , Ai ) be the arborescence which we get from Ai after the α-th step (i < κ, α ≤ λ).
Let us make precise the idea above. Let
A0i = Ai (i < κ).
S
β S
β
α def
If α < λ is a limit ordinal, then Ai = ( β<α Vi ,
β<α Ai ).
β
If α = β + 1, where β < λ is a limit ordinal and the arborescences {Ai }i<κ satisfy
β
condition (1) then add vβ to all of the arborescences {Ai }i<κ which missed it by using
α
lemma 3.5 repeatedly. Denote the resulting arborescences by Ai (i < κ).
β+1
If α = β + 2 where β < λ is an arbitrary ordinal and the arborescences {Ai
}i<κ
S
β+1
β
β def
satisfy condition (1), and the set N
= {vβ+1 } ∪ i<κ Vi
\ Vi is nite, then add
β+1
β
the points of N one by one to all of the arborescences {Ai
}i<κ that missed it, by
α
using lemma 3.5 repeatedly. Denote the resulting arborescences by Ai (i < κ).
Let
Proposition 3.14.
Proof:
doable.
The transnite recursion above does not stop before the λ-th step.
Suppose, seeking a contradiction, that it does.
The limit steps are always
At successor steps we do not violate condition (1) and we extended the
arborescences with only nitely many new points and edges, so if a successor step
is doable then the successor of it is also.
Hence the rst step where the recursion
can not be continued is necessarily a successor of a limit ordinal β . But then β
β
is the rst ordinal such that the arborescences {Ai }i<κ violate condition (1). Let
EGRES Technical Report No.
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Section 4. Conjectures
S
sβ (X) = |{i : Viβ ∩ X = ∅}|, ∅ 6= Y ⊆ V arbitrary and Dfβ = (V, A \ i<κ Aβi ). If
inDf (Y ) = inDβ (Y ), then %Dβ (Y ) = %Df (Y ) ≥ s(Y ) ≥ sβ (Y ). If e ∈ inDf (Y ) \ inDβ (Y )
f
f
f
β
then there is an i0 such that e ∈ Ai . Let γ < β be the smallest ordinal such that
0
(i < κ),
e ∈ Aγi0 . Then γ is a successor ordinal so by the recursion we get end(e) ∈ Aγ+1
i
β
thus end(e) ∈ Ai (i < κ), hence % β (Y ) ≥ 0 = sβ (Y ). So there is no violating Y for
Df
β
the arborescenses {Ai }i<κ which is a contradiction.
Finally the arborescences
4
{Aλi }i<κ
are the desired spanning-arborecences.
Conjectures
4.1
A condition about paths
We show by a counterexample that the condition of niteness of the sets
{i < κ : v ∈
/ Vi }
Hv =
is necessary in theorem 3.1 and formulate a conjecture with a con-
dition about paths (which is a strengthening of the condition (1)) motivated by this
counterexample.
D (see gure 2) and a system of subarborescences of it such
that Df does not contain even 5-length paths and the system satises condition (1)
but the desired extensions of the arborescences do not exist. Let D = (V, A) where
V = {rn }n∈N ∪{v}, |eD (r0 , rn )| = ℵ0 (n ∈ N+ ), |eD (rn , v)| = 1 (n ∈ N+ ), |eD (v, r0 )| =
ℵ0 and An = ({rn }, ∅) (n ∈ N). Observe that if P is a path of length 3 in D then r0
and v must belong to, hence Df (= D) does not contain a path of length 3. We show
that the system above does not violate condition (1). Let ∅ 6= X ⊆ V be arbitrary.
Assume rst r0 ∈
/ X . If rn ∈ X for some n ∈ N+ then %Df (X) = ℵ0 , if not then
X = {v} and %Df (X) = ℵ0 again. Assume r0 ∈ X . If v ∈
/ X then %Df (X) = ℵ0 . Else
if each elements of {rn }n∈N+ \ X has one-one outgoing edge to X so there is equality
We construct a digraph
in condition (1).
Otherwise, obviously there is no system of edge-disjoint paths
goes from
rn
to
v
thus we can not extend the
An -s
{Pn }n∈N
such that
Pn
to edge-disjoint spanning arbores-
cences.
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4.2
9
An arborescence-analog of the Erd®s-Menger theorem
r1
r2
r3
r0
v
rn
Figure 2: The counterexample.
(Thick arrows stand for countably innite parallel
edges).
Conjecture 4.1. Let D = (V, A) a digraph, Ai = (Vi , Ai ) (i < κ) edge-disjoint
subarborescences of D and Df = (V, A\∪i<κ Ai ). Suppose Df does not contain forwardinnite paths and for all v ∈ V there is a system of edge-disjoint paths {Pi }i<κ in Df
such that Pi goes from Vi to v . Then there are extensions of the Ai -s to edge-disjoint
spanning arborescences of D without changing their roots.
4.2
An arborescence-analog of the Erd®s-Menger theorem
D = (V, A) is a nite
digraph with distinct vertices s, t then the maximum number of the edge-disjoint s − t
def
paths is equal to λD (s, t) = min{%D (X) : t ∈ X ⊆ V \ {s}}. If D is innite but
λD (s, t) is still nite then Menger's theorem remains true, actually it is derivable from
the nite version. Let D be an innite digraph and κ = λD (s, t) ≥ ℵ0 . Erd®s observed
The edge-version of the directed Menger's theorem states that if
that the theorem above can be extended to this case easily in the form that there is a
system
{Pi }i<κ
of edge-disjoint
s−t
paths. He also realized that his generalization is
S{Pi }i<κ of edge-disjoint s − t
inD (X) ⊆ i<κ Ai and |Ai ∩ inD (X)| = 1
too rough and conjectured that there is also a system
paths and a set
for all
i < κ.
t ∈ X ⊆ V \ {s}
such that
It was called the Erd®s-Menger conjecture and after a sequence of partial
results was decided armatively by R. Aharoni and E. Berger in [8].
An analog of Erd®s' generalization above is true in the context of arborescences.
Namely let
κ
D
be
κ-edge-connected
from the vertex
edge-disjoint spanning arborescences rooted at
r.
r,
where
κ ≥ ℵ0 .
Then
D
has
The proof is not too hard, the
main idea is that one can build the desired spanning arborescences by an almost
greedy trannite recursion taking care only of the property that after certain steps
the pointset of the initial arborescences should be the same.
In the spirit of the
Erd®s-Menger theorem we formulate a strengthening of the fact above.
Let D = (V, A) be a digraph and κ ≥ ℵ0 its edge-connectivity from r,
then there is a system of edge-disjoint spanning arborescences
Ai = (V, Ai ) (i < κ) of
S
D rooted at r and ∅ 6= X ⊆ V \{r} such that inD (X) ⊆ i<κ Ai and |Ai ∩ inD (X)| = 1
holds for all i < κ.
Conjecture 4.2.
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10
References
Note that inniteness of the cardinal
κ
can not be omitted by [7].
References
[1] J. Edmonds:
Edge-disjoint branchings
In B. Rustin, editor, Combinatorial Algo-
rithms, pages 9196, Academic Press, 1973.
[2] L. Lovász:
On two minimax theorems in graphs,
J. Comb. Theory, Ser. B 21(2):
96-103 (1976).
[3] N. Kamiyama, N. Katoh and A. Takizawa:
Graphs,
Combinatorica, 29(2):
Arc-disjoint In-trees in Directed
197-214, 2009. 19th Annual ACM/SIAM Sym-
posium on Discrete Algorithms (SODA), 518-526, 2008.
[4] S. Fujishige:
A note on disjoint arborescences.
Combinatorica, 30(2):
247-252,
(2010).
[5] K. Bérczi and A. Frank:
Packing Arborescences, Combinatorial Optimization and
Discrete Algorithms, RIMS Kokyuroku Bessatsu 23: 131 (2010).
[6] Cs. Király:
2013-03
On maximal independent arborescence-packing. Technical Report TR-
(https://www.cs.elte.hu/egres/www/mf_techrep.html),
Egerváry
Research Group, (March 2013).
[7] R. Aharoni and C. Thomassen:
disjoint spanning trees,
Innite, highly connected digraphs with no two arc-
Journal of Graph Theory, Vol. 13, No. 1: 71-74, (March
1989).
[8] R. Aharoni and E. Berger:
Menger's theorem for innite graphs, Inventiones Math-
ematicae 176: 1-62 (2009)
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