Planar graph

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Transcript of Planar graph

  • 2016/06/28

    2.5

    D3

  • 2016/06/28

    Abstract

    G | | = | | - | | + 2

    G K5 K3,3

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    Outline

    ( )

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    Outline

    ( )

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    2.28

    (simple Jordan curve):

    : [0,1] 2 (= )

    (0) (1)

    (closed Jordan curve):

    (0)=(1) 0x

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    (polygonal arc):

    (polygon):

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    (connected region)

    :

    J R = 2 \ J

    p,q R (= J

    )

    J

    1 (= J )

    1

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    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)}

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    (planar graph)

    G (planar) :

    G

    (face) :

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    K4

    K5 ( )

    ( 24)

    ( )

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    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

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    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

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    2.30

    :

    2. J 2

    p2\J, qJ

    p q 1

    q

    2 J

    J 2\J 2

    2 J J

    2\J 1

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    2.30

    :

    3. J 1

    J 2\J 2

    2 J J

    2\J 1

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    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

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    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

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    (outer face)

    1

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    Outline

    ( )

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    2.31 2-

    :

    1. G ( 2.30)

    2. G

    1

    x x-1

    G 2-

    2

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

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    2.31 2-

    :

    1. G ( 2.30)

    - G {Je}

    2

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

    G 2-

    2

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

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    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

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    2.32

    :

    1. G 2- ( 2.31)

    2.

    G

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

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    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

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    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

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    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

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    Outline

    ( )

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    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

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    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

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    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

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    2.37 33-

    K5 K3,3:

    )

    )

    2.36 e={v,w} 3-

    ( ) 2-

    ( 2.31)

    K5 K3,3

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    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

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    2.37 33-

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

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

    K5

    wv

    y1y2

    y3

    y4y5

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    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

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    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

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    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

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    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

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    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

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    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

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    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})

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    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

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    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]

    ( )