Proof of the Doubling Lemma for Trees

This subsection concerns the proof of Lemma5.7. The proof uses the same ideas as the proof of Theorem5.1 to find the cut (B, W) or the cut (B, W, Z) with the additional setZ. Again, aP-labeling of the vertices for a longest pathP will be used and some vertices inP will be called special. However, since the aim is to find anm-cut and not a bisection, the bijectionNm(v) =v+m, which is defined as in the proof of Lemma5.3, is used instead of the bijectionA(v) =v+¹₂n. As discussed after the proof of Theorem5.1, some ideas used in the proof of Theorem 5.1 do not work for the bijection Nm(v) for general m. To generalize these ideas, a backward and a forward version of special vertices is defined, which also leads to backward and forward versions of the setsPv andHv. Table5.1provides an overview on the notation used in the proof of Theorem5.1and Lemma5.7.

LetG= (V, E) be an arbitrary forest onnvertices and fix somem∈[n]. If ∆(G)≤2, then Lemma5.2b) implies that a cut (B, W) satisfying Option1)exists. So assume that ∆(G)≥3. In this case, Lemma5.2a) implies that we may assume thatG is a tree. Setd := diam^∗(G) and letP = (VP, EP) be a longest path inG. Note that |VP|=dn. As ∆(G)≥3, the pathP consists of at least three vertices and has two distinct leaves, which are denoted byx₀ andy₀. LetG⁺ be the graph obtained fromGby inserting the edge{x0, y0} and note that the vertices inVP induce a cycle inG⁺. For technical reasons, we will sometimes refer to the pair{x₀, y₀}as an edge of G, even though it is not. For each vertex v∈VP, letTv

be the component ofG−EP that contains vand defineTv⁰ :=V(Tv)\ {v}. Consider aP-labeling of the vertices ofG, wherex₀receives label 1, and identify each vertex with its label. From now on, any number that differs from a label in [n] by a multiple ofn is considered to be the same as this label. For three verticesa, b, c∈V witha6=c, we say thatbis betweenaandcifb=a,b=c, or if starting ataand going along the numeration given by the labeling reachesbbeforec. Ifa=c, then we say thatbis betweena andc ifb=a=c. For example, whenn= 10, then 5 is between 1 and 7, and 9 is between 8 and 3. For a vertexv∈VP, the unique vertexw∈VP with the property thatTwcontains the vertexv+ 1 is called the vertex afterv on P. Furthermore, ifwis the vertex afterv onP, then the edge {v, w}is referred to as theedge after v on P. Similarly, in this case, the edge {v, w}is called theedge beforewon P andv is called thevertex beforew onP. For each vertex v∈V, the vertexNm(v) :=v+mis called them^th-next vertex of v. Note thatNm:V →V is a bijection and, hence, its inverse functionN_m⁻¹ is well-defined. For a setU ⊆V, defineNm(U) :={Nm(u): u∈U}andN_m⁻¹(U) :=

N_m⁻¹(u): u∈U . Case 1: There is a vertexv∈VP withNm(v)∈VP.

Letv∈VP be a vertex withNm(v)∈VP and letBbe the set of vertices betweenv+ 1 andNm(v), which satisfies|B|=m. Moreover, letW :=V \B andZ:=∅. The cut (B, W) cuts at most two edges. More

Common notation in the proof of Theorem5.1and the proof of Lemma5.7 G= (V, E) forest onnvertices, may assume thatGis a tree,

d:= diam^∗(G),

P = (VP, EP) longest path inG, ends x₀ andy₀ ⇒ |VP|=dn, consider aP-labeling of the vertices, identify each vertex with its label, Tv= components ofG−EP, T_v⁰ =V(Tv)\ {v}.

Table 5.1:Overview of the notation used in the proofs of Theorem5.1and Lemma5.7.

precisely, if they exist, the cut (B, W) cuts the edge afterv onP and the edge afterNm(v) on P, except when m=nand no edge is cut. See also Figure5.2, which refers to Lemma5.3where the same situation arose. Hence, the cut (B, W, Z) satisfies Option 1).

Case 2: There is no vertexv∈VP withNm(v)∈VP.

In this case, Nm(v)6∈VP andN_m⁻¹(v)6∈VP for allv ∈VP. Then, |VP| =|Nm(VP)| ≤ |V \VP|, which implies that|VP| ≤ ¹₂nand

d≤ ¹₂. (5.5)

A vertex v ∈VP is called b-special, if there is a vertexw ∈Tv⁰ withNm⁻¹(w)∈VP. We use b as in backward, because there is some vertexwinTvsuch that goingmsteps backward fromwin the numeration gives a vertex onP. A vertexv∈VP is calledf-specialif there is a vertexw∈Tv⁰ withNm(w)∈VP. We

x Nm⁻¹(˜v)

N_m⁻¹(v)

˜

v v

v^b Nm⁻¹(x)

Tv

Pv^b

Hv^b

. . . . . .

a)A b-special vertexvand the setsP_v^b andH_v^b.

x

Nm(˜v) Nm(v)

˜

v v v^f Nm(x)

Tv

Pv^f

Hv^f

. . . . . .

b)An f-special vertexvand the setsP_v^f andH_v^f.

Figure 5.5:Notation used in the proof of Lemma5.7. A treeTuwithu∈VP is colored blue if the vertexuis b-special. A treeTuwithu∈VP is colored red if the vertexuis f-special.

use f as inforward, because there is some vertexwin Tv such that goingm steps forward fromwin the numeration gives a vertex onP.

For every vertexv∈VP, define

P_v^b:=N_m⁻¹(T_v⁰)∩VP and P_v^f :=Nm(T_v⁰)∩VP.

See Figure5.5for an example. Note that, for everyv∈VP, whenP_v^b6=∅, then the vertices inP_v^b appear consecutively on the unique cycle inG⁺ andPv^b induces a path or a cycle in G⁺. Moreover,Pv^b is not empty if and only ifv is b-special. Furthermore,P_v^b andP_w^b are disjoint for distinct verticesv, w∈VP

and every vertex inP is contained in a setPv^b for some b-specialv∈VP. The same holds analogously for the setsP_v^f and the following proposition is obtained.

Proposition 5.9.

a)

P_v^b: v∈VP is b-special is a partition ofVP. b)

P_v^f: v∈VP is f-special is a partition ofVP.

For each b-specialv∈VP, definev^b=Nm⁻¹(x), wherexis the smallest vertex inTv⁰ with Nm⁻¹(x)∈VP.

Note that, for a b-special vertexv∈VP, if the setPv^b does not consist of all vertices inVP, thenv^b is one of the ends of the path thatP_v^b induces inG⁺. Similarly, for each f-specialv∈VP, definev^f =Nm(x) where xis the smallest vertex in Tv⁰ withNm(x)∈VP. Then, for each f-specialv ∈VP for which Pv^f

does not induce a cycle in G⁺, the vertex v^f is one of the leaves of the path that P_v^f induces inG⁺. Furthermore, let

H_v^b:= [

x∈P_v^b\{v^b}

T_x⁰ for all b-specialv∈VP, H_v^f := [

x∈Pv^f\{v^f}

T_x⁰ for all f-specialv∈VP,

andH_v^b=∅for everyv∈VP that is not b-special as well asH_v^f=∅for everyv∈VP that is not f-special.

Lemma 5.10.

a) For every b-specialv∈VP, the vertexv^b is f-special.

b) For every f-specialv∈VP, the vertexv^f is b-special.

c) A vertexw∈VP is b-special if and only if there exists an f-special vertexv∈VP such thatw=v^f. d) A vertexw∈VP is f-special if and only if there exists a b-special vertex v∈VP such that w=v^b. Proof.

a) Consider a b-special vertexv∈VP and let ˜v∈VP be the vertex beforev on the path P. As the vertexN_m⁻¹(˜v) is not in VP by the assumption of Case 2,N_m⁻¹(˜v) must be inT_v⁰b, i. e.,Tv^b contains the vertexx:=Nm⁻¹(˜v),x6=v^b, andNm(x) = ˜v∈VP. Hence,v^b is f-special. See also Figure5.5a).

b) Consider an f-special vertexv∈VP and let ˜v∈VP be the vertex beforev on the pathP. As the vertexNm(˜v) is not inVP by the assumption of Case 2,Nm(˜v) must be inT_v⁰f, i. e.,Tv^f contains the vertexx=Nm(˜v),x6=v^f, andNm⁻¹(x) = ˜v∈VP. Hence,v^f is b-special. See also Figure5.5b).

c) First, if there is an f-special vertex v∈VP such thatw=v^f, then Partb)shows thatwis b-special.

Second, as two distinct b-special verticesv andwhave distinct verticesv^b andw^b and similarly for f-special vertices, it follows that

w∈VP: wis b-special ≤ w^b∈VP: wis b-special

Parta)

≤ v∈VP: v is f-special ≤ v^f ∈VP: vis f-special

Partb)

≤ w∈VP: wis b-special .

Note that all inequalities in the above equation must be equalities, which shows that a vertexw∈VP

can only be b-special if there exists an f-special vertexv∈VP such thatw=v^f.

d) Analog to Partc). 2

Lemma5.10c)immediately implies that X

v∈V_P: vis f-special

|T_v⁰f| = X

w∈V_P: wis b-special

|Tw⁰| (5.6)

and Lemma5.10d)implies that

X

v∈VP: vis b-special

|T_v⁰b| = X

w∈VP: wis f-special

|T_w⁰|. (5.7)

For every b-specialv∈VP and for every vertexu∈Pv^b∪Hv^b, the vertexNm(u) lies inTv⁰. Thus,

Now, (5.6) and (5.7) allow to rearrange the terms in the sums in the first expression, such that both sums use|Tv⁰|instead of|T_v⁰b|and|T_v⁰f|, respectively. Thus,

As there is at least one b-special vertex v ∈VP and at least one f-special vertex v∈VP, the previous equation implies the following proposition.

Nm⁻¹(˜v)

Nm⁻¹(v)

˜

v v

v^b v^b_`

Tv

Pv^b

Hv^b

. . . . . .

Z

B1

B2

Figure 5.6:Proof of Lemma5.7. Cut (B, W, Z) in Case 2a).

Case 2a) There is a b-specialv∈VP with|Tv⁰|+|Hv^b| ≤ ¹_d−1

|Pv^b|.

Replacing|T_v⁰|with (5.8) yields

|P_v^b|+ 2|H_v^b| ≤ ¹_d−1

|P_v^b| ⇒ 2|P_v^b|+ 2|H_v^b| ≤ ¹_d |P_v^b|

⇒ |P_v^b|+|H_v^b| ≤ ₂¹_d |P_v^b|. (5.14)

Next, a cut (B, W, Z) with the properties required for Option2)in Lemma5.7is constructed. DefineZ :=

H_v^b∪P˙ _v^b, which satisfiesZ6=∅sincevis b-special andP_v^bhence contains at least one vertex. Furthermore,Z andT_v⁰ are disjoint. Indeed, assume that there was a vertexxinZ∩T_v⁰. AsP_v^b⊆VP andT_v⁰ does not contain any vertex fromVP, it follows that x∈H_v^b. By the definition ofH_v^b, the vertexv must be inP_v^b andv 6=v^b. Now, v ∈P_v^b =N_m⁻¹(T_v⁰)∩VP implies that T_v⁰ contains a vertex y such thatN_m⁻¹(y) =v.

Now, any vertex z∈T_v⁰ that is smaller thany satisfiesN_m⁻¹(z)∈T_v⁰. Therefore,v must bev^b, which is a contradiction and implies thatZ andT_v⁰ are disjoint. Together with (5.8) it follows that|Z| ≤ ¹₂n.

Furthermore, (5.14) implies that|Z| ≤ |H_v^b|+|P_v^b| ≤ _2d¹|P_v^b|. AsP_v^b induces a collection of paths inG[Z]

with at most one path in each component ofG[Z], the relative diameter ofG[Z] can be estimated by diam^∗(G[Z]) ≥ 1

|Z||Pv^b| ≥ 2d.

Definev_{`</sub><sup>b</sup>=Nm<sup>−1</sup>(x), wherexis the largest vertex among all vertices inTv<sup>0</sup> withNm<sup>−1</sup>(x)∈VP. Note that, ifP<sub>v</sub><sup>b</sup>6={v<sup>b</sup>}andP<sub>v</sub><sup>b</sup>6=VP, thenv<sub>`}^b is the leaf of the path induced byP_v^b inG⁺that is notv^b. Let ˜v be the vertex beforev onP. Ifv_`^b= ˜v, then defineB1:=∅. Otherwise, define

B₁:=

x∈V: xis betweenv^b_`+ 1 and ˜v ,

see Figure 5.6. Moreover, define ˜m:= m− |B1|. Since Nm(v_`^b) is in Tv⁰, it follows that 1≤m˜ ≤ |Tv⁰|.

Definec:= 2−₁₋¹_d = ¹⁻²_1−d^d and note thatc∈[0,1) by (5.5). Corollary4.4 implies that the forestG[Tv⁰] allows ac-approximate ˜m-cut (B2, W2) of width¹

eG[Tv0](B2, W2) ≤ 2c

1−c

∆(G) = 2(1−2d) d

∆(G)

1Note that Lemma4.5b)provides a better bound on the width of ac-approximate ˜m-cut inG[T_v⁰]. For now, we will use Corollary4.4as the computations are easier. The improvements will be discussed in Section5.2.3.

Nm(˜v) Nm(v)

˜

v v v^f v^f_`

Tv

Pv^f

H^fv

B1

Z B2

. . . . . .

Figure 5.7: Proof of Lemma5.7. Cut (B, W, Z) in Case 2b).

= 2 d−4

∆(G) ≤ 2

d−3

∆(G). (5.15)

Note thatB2∪˙ W2=Tv⁰ andcm˜ ≤ |B2| ≤m.˜

DefineB:=B₁∪B₂ and note thatB₁ andB₂ are disjoint by construction. Moreover, m− |B| = (m− |B₁|)− |B₂| ≤ m˜ −cm˜ ≤ (1−c)|T_v⁰|

≤ d

1−d|T_v⁰| ≤ |P_v^b| ≤ |Z|,

where the second to last inequality holds by the assumption|T_v⁰|+|H_v^b| ≤ ¹_d−1

|P_v^b|of Case 2a). Hence,

|B| ≤m≤ |B|+|Z|. SinceB2⊆T_v⁰ and the setsT_v⁰ andZare disjoint as argued above, the setsB andZ are disjoint.

LetW :=V \(Z∪B). Next, the width of the cut (B, W, Z) is estimated. At most ∆(G) + 1 edges ofG are cut by (Z, B∪W), i. e., at most all edges incident tov^b and the edge after v^b_` onP. Moreover, at mosteG[Tv0](B2, W2) + ∆(G) edges are cut by (B, W) in G[B∪W], which are the edges cut by (B2, W2) withinG[Tv⁰] and all edges incident tov (ifv /∈Z, i. e., ifv6=v^b). With (5.15) it follows that

eG(B, W, Z) ≤ eG(Z, B∪W) +e_G[B∪W](B, W) ≤ ∆(G) + 1 +e_G[T0

v](B2, W₂) + ∆(G)

≤ 3∆(G) +2 d−3

∆(G) ≤ 2 d ∆(G).

Case 2b)There is an f-specialv∈VP with|Tv⁰|+|Hv^f| ≤ ¹_d−1

|Pv^f|.

This case is similar to Case 2a) but not completely analogous as we cannot simply reverse the labeling to obtain the situation of Case 2a), because this does not yield a P-labeling. Analogously to Case 2a),|P_v^f|+|H_v^f| ≤ _2d¹|P_v^f|can be derived by replacing|T_v⁰|with (5.9). Furthermore, defineZ :=H_v^f∪˙ P_v^f and deduce analogously to Case 2a) that diam^∗(G[Z])≥2dand 0<|Z| ≤ ¹₂n, asZ andT_v⁰ are disjoint.

Note thatv is betweenNm⁻¹(v^f) andv^f. Moreover,v=v^f is possible and, in this case,v lies inZ, and otherwisev does not lie inZ. Therefore, the setB1 is defined in a slightly different way than in Case 2a).

Let

B1:=

x∈V: xis betweenv andv^f,x6=v^f

0.2 0.4 0.6 0.8 1 10

20 30 40

50 ₈

d (Theorem5.6)

12

log₂ ¹_d2

+ 7 log₂ ¹_d

+ 6

(Theorem5.12)

relative diameterd paths

stars

Figure 5.8: Comparing the bounds provided by Theorem5.6 and Theorem5.12. The factor ∆(G), which is present in both bounds, is ignored.

and ˜m:=m− |B₁|. See also Figure5.7for a visualization. SinceN_m⁻¹(v^f)∈T_v⁰ andv∈B₁orv=v^f ∈Z, it follows that|Tv⁰| ≥m. Let˜ c:= 2−₁₋¹_d as in Case 2a). Analogously to Case 2a), Corollary4.4implies that the forestG[Tv⁰] admits a c-approximate ˜m-cut (B2, W₂) of widthe_G[Tv0](B2, W₂)≤ ²_d−3∆(G).

Defining B := B1∪˙ B2, one can argue that |B| ≤ m ≤ |B|+|Z| and that B is disjoint from Z. Let W :=V \(B∪Z). Then, at most ∆(G) + 1 edges are cut by (Z, B∪˙ W), i. e., the edges incident tov^f and the edge after the other end v^f_` of Pv^f. At most eG[Tv0](B2, W2) + ∆(G) edges are cut by (B, W) in G[B ∪W], which are the edges cut by (B2, W₂) within G[Tv⁰] and all edges incident to v. All together,eG(B, W, Z)≤ ²_d∆(G) is obtained.

Im Dokument On the Minimum Bisection Problem in Tree-Like and Planar Graphs  (Seite 165-172)