Feasible dijoins

is nested. We call a dicut B ∈B^⊕ nestedly F-tight if B =^LB for some nested setB ⊆ Bof F-tight dibonds.

Lemma 6.7.10. Every finite F-tight B ∈B^⊕ is nestedly F-tight.

Proof. Let B be a decomposition of B into finitely many F-tight dibonds such that the number of pairs (A, A⁰)∈ B × B such thatA andA⁰ cross is minimal. We show that B is nested. Suppose for a contradiction that A, A⁰ ∈ B are crossing.

Then

B⁰ := ((Br{A, A⁰})∪ {A∧A⁰, A∨A⁰})

is a decomposition of B into F-tight dibonds in B by Lemma 6.7.1. An easy calculation shows that every dibond which is nested with both A and A⁰ is also nested with bothA∧A⁰ andA∨A⁰. Similarly, a dibond which crosses bothA∧A⁰ andA∨A⁰ also crosses bothAandA⁰. HenceB⁰ contradicts the minimality choice of B.

As above, the question whether this is in reality a stronger condition remains open.

Question 6.7.11. Is every F-tight dicut B ∈B^⊕ also nestedly F-tight?

(E1) for every finite dicut δ⁻_D(X)∈B^⊕ the equation

|Z∩δ_D⁻

F(X)|=|Z ∩δ_D⁺

F(X)|<∞ holds;

(E2) for every finite dicutδ_D⁻(X)∈B^⊕ with|J(Z, X)| ≥2 there is an F-tight di-bondδ⁻(Y)∈Bsuch that J(Z, X∩Y) andJ(Z, X∪Y) partition J(Z, X).

Remark 6.7.12. Every directed cycle C inDF for which every jumping edge it contains is basic is exchangeable.

We continue with a lemma that allows us to modify a givenB-dijoin to another B-dijoin given some exchangeable set.

Lemma 6.7.13 (Exchange Lemma). If Z ⊆E(D_F) is exchangeable, then FZ := (F r{f ∈F |f^∗ ∈(Z∩F^∗)})∪(Z∩E(D))

is a B-dijoin of D.

Proof. By property (E1), it suffices to show that |δ_D⁻(X)∩F| ≥ |J(Z, X)|+ 1 for each dicut δ_D⁻(X)∈B^⊕, since then either Z∩δ⁻_D(X) is non-empty or {f ∈F ∩δ_D⁻(X)|f^∗ ∈Z∩F^∗} is a proper subset of F ∩δ_D⁻(X). We show this claim by induction on |J(Z, X)|.

For |J(Z, X)|= 0 this is trivial. For |J(Z, X)|= 1, note that |δ_D⁻(X)∩F|>1 since δ_D⁻(X) is not F-tight.

For |J(Z, X)| ≥ 2 let δ⁻(Y) be as in property (E2). Since B is finite-corner-closed, the corners δ_D⁻(X∩Y) and δ_D⁻(X∪Y) are in B^⊕. Now |J(Z, X∩Y)|

and |J(Z, X∪Y)| are both less than|J(Z, X)| by property (E2). Hence we get

|δ⁻_D(X∩Y)∩F| ≥ |J(Z, X∩Y)|+ 1 and |δ_D⁻(X∪Y)∩F| ≥ |J(Z, X∪Y)|+ 1 by induction. Moreover, sinceδ⁻(Y) is anF-tight dibond we get that|J(Z, Y)|= 0 and |δ_D⁻(Y)∩F|= 1. Finally these inequalities together with Remark 6.2.2 and property (E2) yield

|δ⁻_D(X)∩F|=|δ_D⁻(X∩Y)∩F|+|δ_D⁻(X∪Y)∩F| − |δ_D⁻(Y)∩F|

≥ |J(Z, X∩Y)|+ 1 +|J(Z, X∪Y)|+ 1−1

=|J(Z, X ∩Y)|+|J(Z, X∪Y)|+ 1

=|J(Z, X)|+ 1, finishing the proof.

Before we continue, we fix the following notation. LetC be a directed cycle in any digraph D. If e∈E(D) is an edge both of whose endvertices lie on C, then there exists a unique directed cycle in C+e that contains e. We shall call that cycleC(e).

In order to prove a lemma telling us that we can obtain an exchangeable negative directed cycle from any negative directed cycle in D_F, we need the following technical lemma.

Lemma 6.7.14. Let D be a digraph, let Z ⊆D a directed cycle and let M ={s_it_i ∈E(Z)|i∈[m]}a set of m≥2edges such thats_it_i+1 ∈E(D)for every i∈[m] (where t_m+1 :=t₁). Then there exists an integer q >0 such that each edge of E(Z)rM lies in precisely q many of the directed cycles of {C(s_it_i+1)|i∈[m]}.

Proof. Let C := {C(s_it_i+1)|i∈[m]}. We prove the statement by induction on m. For m= 2 we immediately see that q= 1 holds. Now suppose the statement holds for some integer m≥2 with any digraph and any cycle that satisfy the conditions. Let a digraph D, a cycle Z and a set M as in the statement be given such that |M|=m+ 1. We define the auxiliary digraph D⁰ := (D− {s₁t₂, s₂t₃}+s₁t₃)/s₂t₂. Note that in D⁰, the setE(Z)r{s₂t₂}forms a directed cycleZ⁰ containing the m edges of the setM⁰ :=M r{s₂t₂}. Applying the induction hypothesis to M⁰ together with D⁰ andZ⁰ yields some integer q⁰ >0 as in the statement of the lemma.

Next we again consider D,Z andM, and distinguish two cases: If the directed s₁–t₃ path onZ containss₂t₂, then every edge ofE(Z)rM lies in precisely q⁰+ 1 many directed cycles of C. If the directed s₁–t₃ path on Z does not contain s₂t₂, then each edge ofE(Z)rM lies in preciselyq⁰ many directed cycles ofC. Since we get in both cases an integer as required, this competes the proof of the lemma.

The following lemma uses the same ideas as [20, Lemma 9.7.13], including Lemma 6.7.14, whose proof has been omitted in [20, Lemma 9.7.13]. We include a full proof for the sake of completeness.

Lemma 6.7.15. For every negative directed cycleC inD_F there is an exchangeable negative directed cycle Z of DF with ZrJF ⊆CrJF.

Proof. Let C be a negative directed cycle in D_F. If there is a chorde of C in J_F such that C(e) is still negative and strictly smaller than C, then we consider

this cycle instead. Iterating this process yields a negative directed cycle Z with ZrJ_F ⊆CrJ_F such that no further such chord exists. We claim that Z is exchangeable.

Since Z is a finite directed cycle it trivially satisfies property (E1). For prop-erty (E2) we consider the following claim.

Claim. There is an enumeration x₁y₁,· · · , x_ny_n of Z∩J_F such that x_jy_i ∈/ J_F for all i, j ∈[n] with 1≤i < j ≤n.

First we show that if this claim is true, then property (E2) holds. Consider a dicut δ_D⁻(X)∈B withxkyk, x`y` ∈J(Z, X) for somek, `∈[n] with k < ` such that x_iy_i ∈/ J(Z, X) for all i with k < i < `. Then applying Lemma 6.7.6 to U :={x<sub>j</sub>|`≤j ≤n}andW :={y_i|1≤i≤k}gives anF-tight dicutδ⁻(Y) with U ⊆V(D)rY and W ⊆Y. By Lemma 6.7.7 we have that y_j ∈V(D)rY for all j with ` ≤j ≤n andxi ∈Y for alli with 1≤i≤k. Hence with this Y we get the desired bipartition of J(Z, X) into J(Z, X∩Y) and J(Z, X∪Y).

In proving the claim we may assume that|Z∩J_F| ≥2 as otherwise the claim holds trivially. Now we consider an auxiliary digraph H on the vertex setZ ∩J_F where there is an edge with tail xy and head x⁰y⁰ if and only if xy⁰ ∈JF. Then the desired enumeration is an enumeration of V(H) with no backwards edges and hence its existence is equivalent to the non-existence of a directed cycle in H. So assume for a contradiction there is a directed cycle (x₁y₁)· · ·(x_m+1y_m+1) in H (with x1y1 =xm+1ym+1). Hence this cycle corresponds to chordsxiyi+1 ∈JF of Z.

By the minimality of Z the cyclesC(x_iy_i+1) are non-negative for all i∈[m]. By applying Lemma 6.7.14 with D_F, Z and the edges x₁y₁, . . . , x_my_m, there exists an integer q >0 such that each edge in Z − {x_iy_i|i∈[m]} lies in precisely q many cycles C(x_iyi+1). Therefore,

n

X

i=1

cF

C(x_iyi+1)=q·cF(Z), contradicting that Z is negative.

Let us call aB-dijoin F feasible if the auxiliary graphD_F together with the cost-functioncF does not contain any negative cycles. Theorem6.2.21and Lemma6.7.5 then imply that for D_F andc_F there always is a feasible integer-valued potentialπ if F is feasible.

As discussed before, Lemmas 6.7.13 and 6.7.15 immediately yield as a corollary that finite weakly connected digraphs do always contain a feasible B-dijoin, since the exchange process strictly decreases the size of the dijoin.

Corollary 6.7.16. For every finite weakly connected digraph D and every finite-corner-closed class B of finite dibonds of D there is a feasible B-dijoin.

The question about the existence of a feasible potential in infinite digraphs is a cornerstone of this proof method. While we conjecture that they always exist, we are not able to prove this conjecture in its entirety. We will prove some special cases of this conjecture in Section 6.8.

Conjecture 6.7.17. For every weakly connected digraphDand every finite-corner-closed class B of finite dibonds of D there is a feasible B-dijoin.

This conjecture is weaker than Conjecture 6.1.3, since as the following lemma illustrates each B-dijoin that features in a B-optimal pair is feasible.

Lemma 6.7.18. Let D be a weakly connected digraph, let B be a finite-corner-closed class of finite dibonds of D, and let (F,B) be a B-optimal pair for D.

Then F is feasible.

Proof. Suppose for a contradiction that D_F contains a negative directed cycle C.

Let F^∗∩E(C) = {f₁^∗, f₂^∗, . . . , f_n^∗} for somen ∈N. Also let f_i ∈E(D) denote the edge that is mapped to f_i^∗ by ^∗ for every i∈[n]. Since (F,B) is a B-optimal pair forD, there exists a unique finite dicut B_i ∈ B containingf_i for everyi∈[n]. Fur-thermore,B_i contains no further edge of F by definition. Hence, every B_i is an F -tight dicut of D. Let X_i, Y_i ⊆V(D) be the sides of B_i such that E_D(X_i, Y_i) =B_i holds for every i∈[n]. Since B_i is F-tight, we know by Lemma 6.7.7 thatB_i is also F-rigid for every i∈[n].

Next let us consider the intersection of C with any cut EDF(X_i, Yi) in DF. Since C is a directed cycle, we know that

|E(C)∩−→

E_DF(X_i, Y_i)|=|E(C)∩−→

E_DF(Y_i, X_i)|

holds. However, since B_i is F-rigid and contains only the edgef_i from F, there exists an edge e_i ∈E(C)∩E_DF(X_i, Y_i) such that c(e_i) = 1 holds for every i∈[n].

Furthermore, we know thate_i 6=e_j holds ifi, j ∈[n] andi6=j, because all elements

of Bare disjoint by definition. Now we have a contradiction to the negativity of C by the following inequality:

c_F(C)≥

n

X

i=1

c_F(f_i^∗) +

n

X

i=1

c_F(e_i)≥0.

The converse of the previous lemma, that every feasible B-dijoin features in a B-optimal pair is not true. We illustrate this fact with the following example.

Example. In this example we shall consider the weakly connected infinite di-graph D depicted twice in Figure 6.7.1. Before we analyse D in detail, let us define Dproperly. Let A={a_i|i∈N}andB ={b_i|i∈N}be two disjoint count-ably infinite sets. Additionally, let r be some set which is neither contained in A nor in B. Now we set

V(D) :=A∪B∪ {r}.

Next, we defineE₁ :={a_ib_i|i∈N}, E₂ :={a_ib_i+1|i∈N},E₃ :={b_ir|i∈N}and E₄ :={b_ib_i+1|i∈N}. Finally, we complete the definition of D by setting

E(D) =E₁∪E₂∪E₃∪E₄.

Now let us first consider the left instance ofD in Figure6.7.1 more closely. The set of grey edges in that instance is F_L:=E₂ and it is easy to check that F_L is a finitary dijoin of D. In particular, F_L is feasible, which follows by checking that the following mapπ_FL :V(D)→Z is a feasible potential:

π_FL(v) =











−1 if v ∈A, 0 otherwise.

Actually, π_FL is even a feasible rooted potential with r as its root. Here we see that the potential threshold τ−1 decomposes into a nested set of disjoint dicuts forming a nested optimal pair for D together with F_L for D. Hence, D is not a counterexample to Conjecture 6.1.5.

Next, let us consider the right instance ofD in Figure 6.7.1. There the set of grey edges is F_R:=E₁∪ {b₀r}. Again it is easy to verify that F_R is a finitary dijoin of D. Furthermore, F_R is feasible, which is witnessed by the following map π_FR :V(D)→Z being a feasible potential:

π_FR(v) =















−2 if v ∈A,

−1 if v ∈B, 0 if v =r.

πF_L = −1 0 0 r

πF_R =−2 −1 0

r

Figure 6.7.1.: Two instances of the digraph D. All edges that are not already directed are meant to be directed from left to right. Each instance of D contains a feasible finitary dijoin F_L andF_R, resp., marked by the grey edges. These dijoins each give rise to a feasible potentialπ_FL and πFR, resp.. However, FR is not part of any optimal pair for D, while F_L is part of some optimal pair for D.

As for π_FL, the map π_FR is even a feasible rooted potential with root r.

Now we shall see that F_R does not feature in any optimal pair for D. In order to see this, consider an edge a_ib_i from F_R for some arbitrary i∈N. The only dibond a_ib_i is contained in is δ⁺({a_i}). Hence, in order for F_R to feature in an optimal pair (F_R,B) for D, we know that δ⁺({a_i})∈ B must hold for every i∈N. However, every finite dicut containing the edge b₀r ∈F_R must also contain an edge a_ib_i+1 for some i∈N. This shows that we cannot find an optimal pair for D in which F_R features as a finitary dijoin of D. Note that this is in contrast to Example 6.3.1, where F_R could feature as a finitary dijoin in an optimal pair, just not in a nested one.

We conclude this subsection with a lemma that will help us prove Conjec-ture 6.7.17 for special classes of digraphs in Section 6.8.

Lemma 6.7.19. LetD be a weakly connected digraph and let Bbe a finite-corner-closed class of dibonds of D. Then there is a B-dijoin F such that for all finite X ⊆V(D) there is a B-dijoin F_X of D such that F ∩E(D[X]) =F_X ∩E(D[X]) and such that D_FX[X] does not contain any negative cycles.

Proof. Without loss of generality we may assume by Lemma 6.4.1 that D is B-separable. We prove this lemma by compactness.

For a finiteX ⊆V(D), letJ_X be the set of allJ ⊆E(D[X]) such that there is a B-dijoinF_X ⊆E(D) ofDwithF_X ∩E(D[X]) =J with the property thatD_FX[X]

does not contain any negative directed cycles. This set is non-empty since by applying Lemmas 6.7.13 and 6.7.15 successively we can eliminate all negative cycles in D_FX[X], as well as finite, since E(D[X]) is finite due to X being finite andDbeingB-separable. Moreover, for finite sets X ⊆Y ⊆V(D) we obtain that J∩E(D[X])∈ JX for eachJ ∈ JY as witnessed by the sameB-dijoinFY ofD. By the compactness principle there is a set F ⊆E(D) such that F ∩E(D[X])∈ J_X for all finite X ⊆V(D). We claim that this F is a B-dijoin as desired.

First note that for any dibond B ∈B there is a finite set X ⊆V(D) with B ⊆E(D[X]). But since the B-dijoin FX witnessing that F ∩E(D[A])∈ JA

meets B, so does F. Thus F is indeed a B-dijoin, finishing the proof.

Im Dokument Connectivity and tree structure in infinite graphs and digraphs (Seite 174-181)