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### Transcript of Planar graph

• 2016/06/28

2.5

D3

• 2016/06/28

Abstract

G | | = | | - | | + 2

G K5 K3,3

• 2016/06/28

Outline

( )

• 2016/06/28

Outline

( )

• 2016/06/28

2.28

(simple Jordan curve):

: [0,1] 2 (= )

(0) (1)

(closed Jordan curve):

(0)=(1) 0x

• 2016/06/28

(polygonal arc):

(polygon):

• 2016/06/28

(connected region)

:

J R = 2 \ J

p,q R (= J

)

J

1 (= J )

1

• 2016/06/28

2.29 : V(G) 2

e = {x,y} Je ( (x), (y) )

e = {x,y}

{Je : eE(G)} G

(Je \ { (x), (y)}) \ { (v) : v 2 V (G)} [

[

e02E(G)\{e}

Je0

!= ;

Je\{(x),(y)}

• 2016/06/28

(planar graph)

G (planar) :

G

(face) :

• 2016/06/28

K4

K5 ( )

( 24)

( )

• 2016/06/28

2.30

:

1. J

2. J 2

2 J

3. J 1

4. J J 2

J 2\J 2

2 J J

2\J 1

• 2016/06/28

2.30

:

1. J

p2\J, qJ p q 2\J{q}

(J J q )

J 2\J 2

2 J J

2\J 1

• 2016/06/28

2.30

:

2. J 2

p2\J, qJ

p q 1

q

2 J

J 2\J 2

2 J J

2\J 1

• 2016/06/28

2.30

:

3. J 1

J 2\J 2

2 J J

2\J 1

• 2016/06/28

2.30

:

4. J 2

p2\J l

l J " " cr(p, l)

cr(p,l) gp() = cr(p,l) mod 2

J 2\J 2

2 J J

2\J 1p J

l

• 2016/06/28

2.30

:

4. J 2

gp() = cr(p,l) mod 2

pspt J ppspt gp gp

pspt J 1 gps, gpt

gp 2 2

J 2\J 2

2 J J

2\J 1

J 2\J 2

2 J J

2\J 1

ps pt

ps pt

l

l

J

J

• 2016/06/28

(outer face)

1

• 2016/06/28

Outline

( )

• 2016/06/28

2.31 2-

:

1. G ( 2.30)

2. G

1

x x-1

G 2-

2

= |E(G)| - |V(G)| + 2

• 2016/06/28

2.31 2-

:

1. G ( 2.30)

- G {Je}

2

{Je} 2 |E(G)| = |V(G)|

G 2-

2

= |E(G)| - |V(G)| + 2

• 2016/06/28

2.31 2-

:

2. G

P G'

F' F' (C )

P x,y C x,y x-y- 2 (Q1, Q2 )

Q1+P, Q2+P F' 2 1

P |E(P)| - (|V(P)| - |{x,y}|) = 1 = -

G 2-

2

= |E(G)| - |V(G)| + 2

P Q1 Q2

• 2016/06/28

2.32

:

1. G 2- ( 2.31)

2.

G

= |E(G)| - |V(G)| + 2

• 2016/06/28

2.32

:

2.

G1, ..., Gk Fk |E(Gk)| - |V(Gk)| + 2 1

+

F = k Fi-1 + 1 = k(|E(Gi)| - |V(Gi)| + 1) + 1 = |E(G)| - (|V(G)| + k - 1) + k + 1 = |E(G)| - |V(G)| + 2

G

= |E(G)| - |V(G)| + 2

Gi x

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(girth)

G

= 4

• 2016/06/28

2.33

:

G r r = |E(G)| + |V(G)| + 2 ( 2.32)

G 2- 2.31 2 (= )

k kr 2|E(G)|

|E(G)| - |V(G)| + 2 2/k |E(G)| |E(G)| (n-2) k / (k-2)

G 2- 2-

k3 (n-2)3 / (3-2) = 3n-6 ( )

G n k 2-

(n-2) k / (k-2)

G n 3 3n-6

• 2016/06/28

2.34 K5, K3,3

:

2.33

K5: 10 > 35 - 6 = 9

K3,3: 2- 4 9 > (6-2)4 / (4-2) = 8

K5, K3,3

K5: 5K3,3: 3 3

• 2016/06/28

Outline

( )

• 2016/06/28

2.35 G,H:

H H' V(H') = V1

... Vk H' Vi 1 G

G H

H (1) (G-v) (2) (G-e)

(3) (G/e) G G H

H G

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K5, K3,3

H G

• 2016/06/28

2.36 35 3- G G/e 3-

e

:

e

e = {v,w} 3-

G - {v,w,x} x C

C x y G/{x,y} 3-

G - {x,y,z} z

v,w v,w D

D C ( ) C

zy

x v wCD

• 2016/06/28

2.36 35 3- G G/e 3-

e

: D C ( )

C: G - {v,w,x}

D: v,w G - {x,y,z}

y dD {y,d} d V(C)

D: d v,w,x,y,z

C: d x,y,z

D C yV(C) yV(D) D C

zy

x v wC

Dd

• 2016/06/28

2.37 33-

K5 K3,3:

)

)

2.36 e={v,w} 3-

( ) 2-

( 2.31)

K5 K3,3

• 2016/06/28

2.37 33-

K5 K3,3:

)

e={v,w} G/e x

G/e-x (x) C

w v y1, ..., yk

C yi-yi+1 Pi w (w) {v}Pi i

( )

w v

P2P1

P3P4

P5

• 2016/06/28

2.37 33-

K5 K3,3: C = Pi (w) {v}Pi i

(1) w y1, ..., yk 3 :

K5

wv

y1y2

y3

y4y5

• 2016/06/28

2.37 33-

K5 K3,3: C = Pi (w) {v}Pi i

(2) w y1, ..., yk 2 :

K3,3

wv

ysyi

ytyj

w v

ys

yt yj

yi

• 2016/06/28

2.37 33-

K5 K3,3: C = Pi (w) {v}Pi i

(3) w Pi Pi :

K3,3

(1)~(3) w

v

yiyi+1

z2w v

yi

yi+1 z2

z1

z1

• 2016/06/28

2.38

G 3- 5 K5 K3,3

e={v,w} G+e K5 K3,3

2 v, w

:

1.

2. x x x y, z

3. 2- x,y

a. 3- ( 2.37)

b. 2

• 2016/06/28

2.38

G 3- 5 K5 K3,3

e={v,w} G+e K5 K3,3

2 v, w

:

1. :

3- G-X X = {x,y}

G1 = G[V(C)X], G2 = G-V(C)

: v,w V(G1) e = {v,w} K5, K3,3 G+e

G1+e+f G2+f K5, K3,3 (f = {x,y})

y

xG2G1

• 2016/06/28

2.38

:

V(G1) Z1, ..., Zt

Zi K5 (t=5) K3,3 (t=6) Z

Zi V(G1) \ X Zj V(G2) \ X i, j

(x, y 2 Zk Zi, Zj K5, K3,3 3- )

(a) Zi V(G1) \ X Zi G2+f K5, K3,3

(b) Zi V(G2) \ X Zi G1+e+f K5, K3,3

: v,w V(G1) e = {v,w} K5, K3,3 G+e

G1+e+f G2+f K5, K3,3 (f = {x,y})

y

xG2G1

f

• 2016/06/28

2.38

G 3- 5 K5 K3,3

e={v,w} G+e K5 K3,3

2 v, w

: 2. x x x y, z

(G 3- )

z G-x y zV(G1)

e = {y,z} G+e K5, K3,3 G1+e

G2 K5, K3,3 (f G )

G1+e y 2 G1 G2 G

: v,w V(G1) e = {v,w} K5, K3,3 G+e

G1+e+f G2+f K5, K3,3 (f = {x,y})

y

xG2G1

fz

• 2016/06/28

2.38

: 3. 2- x,y

G 2- f = {x,y} E(G)

G+f K5, K3,3 G1+f

G2+f K5, K3,3 G 2- G1, G2

x-y- f

G1, G2 K5, K3,3

G 3- 5 K5 K3,3

e={v,w} G+e K5 K3,3

2 v, w

: v,w V(G1) e = {v,w} K5, K3,3 G+e

G1+e+f G2+f K5, K3,3 (f = {x,y})

y

xG2G1

• 2016/06/28

2.38

G 3- 5 K5 K3,3

e={v,w} G+e K5 K3,3

2 v, w

: 3-a. 3- ( 2.37)

Gi 5 K5 K3,3

2.37 Gi 3-

Gi e e OK

: v,w V(G1) e = {v,w} K5, K3,3 G+e

G1+e+f G2+f K5, K3,3 (f = {x,y})

• 2016/06/28

2.38

G 3- 5 K5 K3,3

e={v,w} G+e K5 K3,3

2 v, w

: 3-b. 2

Gi i f = {x,y} Fi

( 31)

zi{x,y} Fi e = {z1,z2} OK

G+e K5 K3,3

y

xF2F1 z2z1

• 2016/06/28

2.38

G 3- 5 K5 K3,3

e={v,w} G+e K5 K3,3

2 v, w

: zi{x,y} Fi e = {z1,z2} OK

G+e Z1, ..., Zt Zi K5 (t=5)

K3,3 (t=6)

(1) V(G1) \ {x,y} Zi 1

w F2 y

xF2

z2w

• 2016/06/28

2.38

G 3- 5 K5 K3,3

e={v,w} G+e K5 K3,3

2 v, w

: zi{x,y} Fi e = {z1,z2} OK

G+e Z1, ..., Zt Zi K5 (t=5) K3,3 (t=6)

(2) V(Gk) \ {x,y} Zi 2

Z1,Z2 V(G1) \ {x,y}, Z3,Z4 V(G2) \ {x,y} z1Z1, z2Z3,

Z1 Z3 K5

Z1 Z3 Z5, Z6 K3,3

yx

Z4

z2

Z2

Z3Z1

z1

• 2016/06/28

2.39 Kuratowski

:

)

) 2.37 2.38

3- 2.38 3-

K5, K3,3 3- 2.37

K5K3,3

• 2016/06/28

Kuratowski

( 29)

K5 K3,3

( 28(b))

3- 3-

K5 K3,3

• 2016/06/28

2.40

Hopcroft and Tarjan [1974]

( )