Upper Bound for the Width of Exact Cuts in Tree-Like Graphs

Here, it is shown that a bisection with the properties in Theorem1.9exists. First, the existence part of Theorem 1.9is restated and generalized tom-cuts.

As in the proof of Theorem5.6, the heart of the proof for the previous theorem is a “doubling lemma”.

There, Lemma5.7 and its improved version Lemma5.13state that either an exact cut of small width exists or a subgraph with at least twice as large diameter as the original graph exists. Here, the relative weight of a path in a tree decomposition is doubled. The next lemma uses the following notation: Consider a tree decomposition (T,X) withX = (Xⁱ)i∈V(T)and a path P ⊆T withP= (i0, i₁, . . . , i`). The endi<sub>0</sub> is called anonredundant end ofP with respect toX ifX<sup>i0</sup> 6=∅and, if` >0,X^ih 6⊆X^ih−1 for all h∈[`].

The pathP is callednonredundant with respect toX if at least one of its endsi₀ori`is a nonredundant end of P with respect to X. Note that, if (T,X) is a nonredundant tree decomposition, then every pathP ⊆T is nonredundant with respect to X.

Lemma 5.22.

For every graphG on n vertices, for every tree decomposition (T,X) of G of width at most t−1, for everym∈[n], and for every pathP⊆T that is nonredundant with respect to X, there is a cut (B, W, Z) inGthat satisfies one of the following options

1) |B|=m,Z=∅, andeG(B, W)≤2t∆(G), or

In Option1)in Lemma5.22, the bound is increased by a factor oft∆(G) compared to Lemma5.13, as Lemma2.16b) is used now instead of cutting single edges. In Option2)in Lemma5.13, cuts resulting from removing a vertex from the tree were applied, which is extended by using Lemma2.16a). Therefore, an extra factor oft in the bound on eG(B, W, Z) in Option 2)in Lemma5.22appears. Observe that, in Lemma 5.22, when choosing P to be a heaviest path in the tree decomposition (T,X), then the tree decomposition (T⁰,X⁰) in Option 2)satisfies r(T⁰,X⁰)≥r(T,X). As in the previous section, the proof of Lemma5.22is long and technical. Therefore, it is postponed to Section 5.3.2and the proof for Theorem5.21is presented first. Before presenting the proof of Theorem5.21, consider the next proposition about the relative weight of a heaviest path when turning a tree decomposition into a nonredundant one.

Proposition 5.23.

For every tree decomposition (T,X) of a graphG, there is a nonredundant tree decomposition (T⁰,X⁰) ofGsuch that

• the width of (T⁰,X⁰) is the width of(T,X)and

• r(T⁰,X⁰)≥r(T,X).

Proof. Let (T,X) withX = (Xⁱ)i∈V(T) be an arbitrary tree decomposition of some graphG. Denote byt−1 the width of (T,X) and letP⊆T be a heaviest path inTwith respect toX, i. e.,w^∗_X(P) =r(T,X).

Assume that (T,X) is not nonredundant. Then, there exists an edge{i, j} ∈E(T) withXⁱ⊆X^j. LetT⁰be the tree obtained fromTby contracting the edge{i, j}and calling the resulting nodeh. If{i, j}∩V(P) =∅, letP⁰=P. If{i, j} ∩V(P) ={i, j}, letP⁰ be the path obtained fromP by contracting the edge{i, j}. If {i, j} ∩V(P) ={i} or{i, j} ∩V(P) ={j}, then letP⁰ be the path obtained from P by renaming i orj, respectively, toh. Furthermore, letX⁰ be the collection of clusters obtained fromX by removing the clustersXⁱ andX^j and inserting the new clusterX^h=X^j. Then, (T⁰,X⁰) is a tree decomposition of G of widtht−1 and P⁰ is a path inT⁰ withw_X0(P⁰)≥w_X(P). Thus, r(T⁰,X⁰)≥r(T,X). If (T⁰,X⁰) is not nonredundant, then the same argument can be repeated until a nonredundant tree decomposition

ofGwith the desired properties is obtained. 2

The next corollary can be seen as a forerunner of Theorem5.21 and is a corollary to Lemma 5.22, similarly as Lemma5.3follows from Lemma5.7as mentioned after Lemma5.7. Note that Corollary5.24 treats a special case of Theorem5.21for which it presents a bound on the cut width that is better by a constant factor.

Corollary 5.24.

For every graphGonnvertices, everym∈[n], and every tree decomposition(T,X)ofGwithr(T,X)>¹₂, there is an m-cut(B, W)inG witheG(B, W)≤2t∆(G), wheret−1 denotes the width of(T,X).

Proof. LetG= (V, E) be an arbitrary graph and fix some integerm∈[n]. Moreover, let (T,X) be a tree decomposition ofGwithr(T,X)> ¹₂ and widtht−1. Due to Proposition5.23, we may assume that (T,X) is nonredundant. LetP be a heaviest path inT with respect toX and note thatP is nonredundant with respect toX as (T,X) is nonredundant.

Recall that, for every tree decomposition (TH,XH) of an arbitrary graphH and for every pathP⊆TH, the relative weight of PH with respect to X_H is at most 1. Now, sincew^∗_X(P) = r(T,X) > ¹₂, there cannot be a pathP⁰ with relative weight at least 2w^∗_X(P) with respect to X⁰, where (T⁰,X⁰) is a tree decomposition of a subgraphG[Z] ofGwithZ 6=∅. Hence, when Lemma5.22is applied to G, (T,X), and the pathP with size-parameterm, Option2)cannot occur. Therefore, Option1)implies that a cut

with the desired properties exists. 2

Proof of Theorem 5.21. LetG= (V, E) be an arbitrary graph onnvertices and fix some integerm∈[n].

Let (T,X) be an arbitrary tree decomposition ofGof width at mostt−1. We will show that Algorithm5.4 produces anm-cut inGwith the desired properties.

First of all, the while loop in Lines5-9eventually terminates as the graphGshrinks in each round due to|Z˜|<|V(G)|. The setB is initialized as the empty set in Line 4and then only modified in Line7. As the set ˜B computed in Line6contains at mostm− |B|vertices, the returned setB will have sizemas required. Before starting to analyze the width of the cut produced by Algorithm5.4, some invariants are stated and it is shown that Algorithm5.4can be carried out.

Let s^∗ be the number of executions of the while loop. Define G0 = G, let (T0,X₀) be the tree decomposition (T,X) after Line1has been executed, let P0 be the path P computed in Line 3, and set B0 = ∅. Observe that these are the states of the corresponding variables before executing the

Algorithm 5.4:Computes anm-cut.

Input: graph G= (V, E) onnvertices, integerm∈[n], and a tree decomposition (T,X) ofG.

Output: anm-cut (B, W) inG.

1 Transform (T,X) into a nonredundant tree decomposition ofGas in Proposition5.23;

2 LetG0 be a copy ofG;

3 Compute a heaviest pathP inT with respect toX;

4 B← ∅;

5 While|B|< mdo

6 Apply Lemma5.22to the graphG, the tree decomposition (T,X), and the pathP, with size-parameter ˜m=m− |B|to obtain a partition ( ˜B,W ,˜ Z) of˜ V(G) as described there;

7 B←B∪˙ B,˜ G←G[ ˜Z];

8 If Option2)occurred during the application of Lemma5.22, update (T,X) to a tree

decomposition (T⁰,X⁰) of Gwithout increasing its width and updateP to a pathP⁰ ⊆T⁰ that is nonredundant with respect to X⁰ and satisfiesw^∗_X0(P⁰)≥2w_X^∗(P);

9 Endw

10 Return (B, V(G0)\B);

while loop for the first time. For each s ∈ [s^∗], denote by Gs the graph G, by (Ts,Xs) the tree decomposition (T,X), by Ps the path P, and by Bs the set B after the s^th execution of the while loop. Furthermore, fors∈[s^∗]∪ {0}, letnsbe the number of vertices of Gsand definews:=w^∗_Xs(Ps).

For s∈[s^∗], denote by ( ˜Bs,W˜s,Z˜s) the partition of the vertex set ofGs−1 computed in Line6during thes^th execution of the while loop. At the beginning of each execution of the while loop, the following invariants hold:

(i) 0< m− |B| ≤ |V(G)|,

(ii) the setB and the vertex set of Gare disjoint,

(iii) (T,X) is a tree decomposition ofGof width at mostt−1, and (iv) P ⊆T is a nonredundant path with respect toX.

Furthermore, for everys∈[s^∗], the following holds:

(v) ws≥2ws−1fors6=s^∗, (vi) ns≤¹₂ns−1 fors6=s^∗, and

(vii) eGs−1( ˜Bs,W˜s,Z˜s)≤t∆(G) log_{2 16}

ws−1

.

By Proposition5.23, the width of (T,X) does not increase when Line1is executed. Furthermore, the pathP computed in Line3is nonredundant with respect toX as (T,X) is nonredundant when Line3is executed. Thus,(i)-(iv)hold before the first execution of the while loop. Assume that(i)-(iv)hold before thes^th execution of the while loop for an arbitrarys∈[s^∗]. It is now argued that thes^th execution of the while loop can be carried out,(i)-(iv)hold before the (s+ 1)^st execution of the while loop ifs6=s^∗, and (v)-(vii) hold for s. By (i), (iii), and (iv), it follows that Lemma 5.22 can be applied with size-parameter ˜m =m− |B|, i. e., Line 6 can be carried out. If Option1) occurs in Line6, then s =s^∗ and(v)-(vii)are satisfied fors, asws−1≤1 implies that 2≤log_{2 16}

ws−1

. If Option2)in Lemma 5.22 occurs in Line6, then (v)-(vii)are satisfied forsby Lemma5.22. In Line6, the vertex set of the graphG is partitioned into the sets ˜Bs, ˜Ws, and ˜Zs with |B˜s| ≤ m− |Bs−1|. By(ii), the union in Line 7 is a disjoint union. Hence, if|B˜s|=m− |Bs−1|, then this is the last execution of the while loop and there is nothing more to prove. So, from now on, assume that|B˜s|< m− |Bs−1|. Then,s < s^∗, Option 2)of Lemma5.22must have occurred, and Line 8is carried out in thes^th execution of the while loop. Hence,

Gis updated toG[ ˜Z] and ˜Bs is added to the setB in Line7, i. e.,Bs=Bs−1∪˙ B˜s. Moreover,

|V(Gs)| = |Z| ≥˜ m˜ − |B˜s| = m− |Bs−1| − |B˜s| = m− |Bs|

and ( ˜Bs,W˜s,Z˜s) is a partition of the vertex set ofGs−1. Therefore,(i)and(ii)hold after thes^th execution of the while loop. Furthermore,(iii)and(iv)are satisfied after thes^th execution of the while loop due to the requirements in Line 8. This completes the proof of the invariants(i)-(vii)and shows that the algorithm can be carried out.

Next, the width of them-cut computed by Algorithm 5.4is analyzed. First, note that in Line1the relative weight of a heaviest path in (T,X) does not decrease by Proposition5.23and, thus,w0≥r(T,X).

With(v), it follows that

which is not possible as the relative weight of a path in a tree decomposition is at most 1. Note that, in the previous equation, the pathPs∗−1 needs to be used as(v)does not apply fors=s^∗. The number of cut edges in the final cut (B, W) is at most the sum of the edges cut in each execution of the while loop, when the vertex set of the current graph is partitioned in Line6. By(vii), this implies that

eG(B, W) ≤ which is larger than the upper bound given in (5.19). Therefore,

eG(B, W) ≤ t∆(G)

ProofofLemma5.7ProofofLemma5.22 forestG=(V,E)onnvertices,assumethatG isatree,graphG=(V,E)onnvertices,treedecomposition(T,X)ofGofwidthatmostt−1, T=(VT,ET),X=(Xi )i∈VT, P=(VP,EP)longestpathinG,endsx0,y0,R=S i∈VPXi,P=(VP,EP)nonredundantpathinTwithrespect toX,endsi0,j0, d=diam∗ (G),|VP|=dn,|R|=rn,r=w^{∗ X}(P), Tv=componentsofG−EP,Ti=componentsofT−EP, v∈VP,Ri={x∈Xi :iisthepath-nodeofx},i∈VP, T0 v=V(Tv)\{v},Si=S j∈V(Ti)Xj\R, labeltheverticesofG,labeltheverticesofG,nolabelingofthenodesofT, Nm(v)=v+m,N−1 m(v)=v−m,Nm(v)=v+m,N−1 m(v)=v−m, v∈VPb-special(f-special)if∃x∈T0 vwith N−1 m(x)∈VP(Nm(x)∈VP),i∈VPb-special(f-special)if∃x∈Siwith N−1 m(x)∈R(Nm(x)∈R), P^{b v}=N−1 m(T0 v)∩VP,Ub i=N−1 m(Si)∩R,Pb i= j∈VP:Rj⊆Ub i, P^{b v}:v∈VPb-specialisapartitionofVP, Ub i:i∈VPb-specialisapartitionofR, Pb i:i∈VPb-specialisapartitionofVP, similarlyforP^{f v}=Nm(T0 v)∩VP,similarlyforUf i=Nm(Si)∩R,similarlyforPf i=n j∈VP:Rj⊆Uf io , vb=N−1 m(x)(vf=Nm(x))forthesmallest x∈T0 vwithN−1 m(x)∈VP(Nm(x)∈VP),ib(if)=path-nodeofN−1 m(x)(Nm(x))forthe smallestx∈SiwithN−1 m(x)∈R(Nm(x)∈R), H^{b v}=S x∈P^{b v}\{vb}T0 x,H^{f v}=S x∈P^{f v}\{vf}T0 x,Hb i=S j∈P^{b i}\{ib}Sj,Hf i=S j∈Pf i\{if}Sj, ∃b-specialvwith|T0 v|+|H^{b v}|≤1 d−1 |P^{b v}| orf-specialvwith|T0 v|+|H^{f v}|≤1 d−1 |P^{f v}|,

∃b-specialiwith|Si|+|Hb i|≤1 r−1 |Ub i| orf-specialiwith|Si|+|Hf i|≤1 r−1 |Uf i|. Z=H^{b v}˙∪P^{b v}orZ=H^{f v}˙∪P^{f v}.Z=Hb i˙∪Ub iorZ=Hf i˙∪Uf i. Table5.2:OverviewonthenotationusedintheproofsofLemma5.7andLemma5.22.Theleftcolumnreferstotheformerone;themiddleandrightcolumn bothrefertothelatterone.