Da Brow Ski Thesis

8/12/2019 Da Brow Ski Thesis

1/153


2/153

P 5


3/153

( K r e )

(K s , K t,t ) A

M 2


4/153

7

(K 3, F ) F

(K 3, S 1,2,3, S 1,1,2 + P 2)


5/153


6/153

H E

A H k k = 3

2C 4 C 4 C 4

S i,j,k H i

C 4

S 1,1,3 C 6 S 1,1,3 C 5 S 1,1,3

S 1,2,3 S 1,1,2 + P 2 Q

G GX X G GX X


7/153


8/153


9/153


10/153


11/153

G V (G) E (G) {x, y}

xy n G m x, y xy E (G)

N (v) v v d(v) v X

N X (v) = N (v) X v X X v U, X

N (U ) = vU N (v) U N X (U ) = vU N X (v) U X

x G N G [x] = N V (G) (x) {x}

d d d d

ab bc cd de bc cd de

G H G H


12/153

V (G) V (H ) E (G) E (H ) G H G H E (G) = E (H ) (V (G) V (G)) U G[U ]

G U U E (G) (U U )

X G G[X ]

X X X

G G G G

G G X Y G (X Y ) \ E (G)

A, B V (G) A B A B A B A

B G M G

V (G) \ M M G G

M M M M = V (G)

K n C n P n n K n,m

n m K n e K n X G G X

G[V (G) \ X ] i , j ,k S i,j,k

i , j ,k i , j ,k S i,j,k

i, j k S 1,1,1 = K 1,3 S 1,2,2

E H

K 1,4 G G G + G G G


13/153

mG m G

H E

H E

d(x, y ) x, y

G G L(G) E (G) L(G)

G G2

G

(G) G G (G)

G (G) G (G) i(G)

D D D

R(r, s ) n n r

s x x x G

G

v i i(v)

G H GH

i j i = j i,j


14/153

i j i j

a,b,c,d,e

3,2(3(e) 32(21(3,2(3(d) 32(21(3,2(3(c) 2,1(2(b) 1(a)))))))))

M = {G1, G2, . . .}

G M G1, G2, . . . M G M H

G H M Free (M ) M Free (M )

Free (M ) C M C C = Free (M )

G X G X G X

G

X

MFIS (X ) X

X X = Free (MFIS (X )) MFIS (X )

G X G X MFIS (X ) G

X G F ree (MFIS (X )) X F ree (MFIS (X )) G Free (MFIS (X ))

G X H G X H = G H MFIS (X ) G

Free (MFIS (X )) G X Free (MFIS (X )) X

MFIS (X ) N X = F ree (N ) MFIS (X ) N H

MFIS (X ) \ N H


15/153

H X Free (N ) H N H Free (N ) H Free (MFIS (X ))

H MFIS (X )

X = Free (M ) Y = F ree (N ) X Y H N G M G

H

Free (C 3, C 4, C 5, . . . )

Free (C 3, C 5, C 7, . . . )

Free (C 4, C 5, 2K 2)

Free (P 4)

P

NP


16/153

G k G k NP

k G

P Q A P Q x

P y Q P (x) = Q(y) =

NP

P = NP

P = N P

k

(G, k ) G k

f (k)nO(1) n G f (k)

k n

P Q A

P Q (x, k ) P

(y, k ) Q P (x, k ) = P (y, k ) = k g(k) g

k k


17/153

W t h

C

t h C

k t 1 W [t]

t h hW [0] W [t] W [t]

W [1] G k W [2] G

k W [0], W [1] W [2] W

(G, k ) (G , k ) k G

k (G , k )

G L(G)

n f () p(n) f () n p(n) n


18/153

X V (G) \ X

G n f ( ) p(n) f ( ) n

p(n) 2

f (2) p(n)

P 5

L(G)2


19/153

F F

K 3

(K 3, F ) F


20/153


21/153

G k G k

G (G)

(G)

W [1]

> 0 n1


22/153

P 5 P 5

P 5 P 5

P 5

S G S G V (G) \ S G

G G G

G L(G) L(G)2


23/153

G (B,W,E ) B W G E B W G

G S S = V (G) \ S S S B S

W S B W |W | > |B | N S (W ) B H = G[W B ] S G H ( H ) = |W | | B |

S H ( H )

T S W = T \ S B = S \ T G[W B ] S H = G[W B ]

S G T = ( S W ) \ B S T S

H

S

G

S

G S G S H H S


24/153

P 5

P 5 P 5

P 5

P 5 (P 5, K 3,z e)

(P 5, K 3,3 e) (P 5, K 2,z )

(P 5, K 2,z e)


25/153

{P 2k+1 : k N} {K k,k +1 : k N}

Ak K 1,k

claw P 4

P 6, C 4

p M p Rb(s, t ) Rb(s, t )

G Rb(s, t ) G K s,s G

K t,t

t p N (t, p) N (t, p ) K t,t

M p


26/153

p = 1 t N (t, p) = 1 t p

p = 2 s r r 1 p = 2 s

s 0 N (t, 2 p) = Rb(t,Rb (t, N (t, p))) G Rb(t,Rb(t, N (t, p)))

G G K t,t

G K Rb(t,N (t,p )) ,Rb (t,N (t,p )) A B C D A

B G A,B,C,D G[A C ] G[B D ]

Rb(t, N (t, p )) A B C D G[C D ] K t,t

K N (t,p ),N (t,p ) C C D D A A B B C D

G G[A B ] G[C D ] G[A C ] G[B D ] N (t, p )

G K t,t M p G M 2 p

C C

C |W | = |B | + 1 C

{P 2k+1 : k N}

{K k,k +1 : k N}

Ak K 1,k

C C t C P t K t,t +1 At

C P t C N (t, t ) + 2


27/153

G = ( B,W,E ) C W W |N B (W )| | W | (N B (W ), W , E (N B (W ) W ))

G M B W W

B |W | = |B | + 1 G = ( B,W,E ) C x

N (t, t )+2 X x M x X N (t, t )

Y M X G[X Y ] N (t, t ) K t,t

t Z G[Z {x}] At At C At C

t (K t,t , P t , At )

i , j ,k

(P i , K j,j +1 , Ak )

(P i , K j,j +1 , Ak ) i , j ,k

P 5

P 5

At

t 2 P t

t 5 K t,t +1 t


28/153

P 5

H x y H N (x) N (y) N (y) N (x)

P 5 P 5

P 5

d1, . . . , d k d1 d2 . . . dk Bk (d1, . . . , d k ) B = {b1, . . . , bk} W = {w1, . . . , w d1 } wi b j i d j Bk(d1, . . . , d k )

P 5 Bk (d1, . . . , d k ) k < d1 d2 . . . dk > 0

H = G[W B ]

S

B

W

H B W B x y W H { x, y}

S H H

Bk (d1, . . . , d k ) = G[W B ]

S

G

G

K 1,2 k > 1 d2 d1 1

k > 1 G[b1, w1, w2] K 1,2 d2 < d 1 1 G[b1, wd1 , wd1 1] K 1,2


29/153

z 4 K 3,z e K 3,z B3(z ,z ,z 1) Bz (3, 3, . . . , 3, 2)

(z 1)

z 4 G (P 5, K 3,z e) S H = Bk (d1, . . . , d k ) = G[W B ]

S

G S 4z + 1

H = Bk ( , , . . . , ) k, K k,

H = Bk ( + 1 , , , . . . , ) k, K k,

G S 4z + 1 K 1,2

d1 > 2z, k 2z d2 d1 1 2z dk > 0 dz z 1 G[b1, . . . , bz , wz , . . . , w 2z ]

2z + 1 dz z dz+ i < d z i > 0 G[b1, b2, . . . , bz 1, bz+ i , w1, w2, wdz ] K 3,z e z 4) dz = dz+1 = = dk di z i {1, . . . , k }

di < d i

1 i 3 G[b1, b2, bi , w1, . . . , w z

1, wdi 1

] K 3,z e d2 = d3 = = dz

d1 d2 d1 1

(P 5, K 3,3 e)

4z+1 G (P

5, K

3,z e) S

G S 4z + 1

S G x y


30/153

x x

K 1,2 2z+1 x 2z + 1

4z + 1 x 4z + 1

z G G G

P 4 P 4 P 5 (K 3,z e)

G C x C

N S (x) N S (C ) C x C (x) C x C (x)

G[C (x)] C x C

C

C C 1 C 2

C

C 1 C 2 C 12 :=C 1 C 2 u C 1 = C 2

C 11 := C 1 \ C 2 v C

C 1 C 2 C 1 x C 2 y

u v C 1 C 11 C 12 C 1

y v v C (y) v C 2 C (y) v C 2

v u y v

x y z u v


31/153

x y G[x,y,v,z ,u ] P 5 x y x y a

b G[x,a,u,b,y ] P 5

C 0 C C C 0

C 0 C C

C S H

H

K k, k, K k,

H

S S H H

H H S W H C

W C 0 W

x y W C 0 z C j

a C C j a z C j C (a) b1, . . . , bz 1 C a

G[a, b1, . . . , bz 1,x ,y,z ] K 3,z e C 0

W W C 0

C 0 W H 4z + 1 W

2z + 1 x,y,z W

C i x W C i , y W C i , z W C i a C C i


32/153

a x a y x y C i C (a) y C (a) y a z a b1, . . . , bz 1 z 1 y z

a G[a, b1, . . . , bz 1,x ,y,z ] K 3,z e

B C 0 C i a C C i

a B C 0 = C 0 B x a H x a a

H H P 5 a1 V (H ) \ V (H ) a a2 N S (a) a3 B \ { x} a4 N S (C i ) \ N S (a) G[a1, a ,a 2, a3, a 4] P 5

x a H

S 4z + 1

2z

z

C i C j N S (C i) N S (C j ) = C i C j

C i C j z 1 C i C j

N S (C j ) N S (C i) K 3,z e x C i y C j

w C j w y G C j

G[C j ] w y a N S (C i) b N S (C j ) G[a,x,w,b,y ] P 5

G S C i C i C j N S (C i )


33/153

N S (C j ) = C i C j N S (C i) N S (C j ) = C i C j

w(C j ) C j C j w(C j ) = (G[C j ]) | N S (C j )| C j

w(C j ) = (G[C j ]) + 1 | N S (C j )| (G[C j ])

Q = {v1, . . . , v p} v I v v

H (v) H B (v) := N S (I v) H W (v) v C i

H W (v) := I v H (v) v C i a C i a

H W (v) := I v {a} H (v) K k, a

H B (v1), . . . , H B (v p) G P 5 pi=1 H

W (vi ) H Q H (vi )

Q Q H Q S Q H Q

(P 5, K 3,z e)

G (P 5, K 3,z e) G

S G

S G

H H 4z+1

V (G) \ S

z

C i S (C i ) = G[C i] | G[C i ])| < z

Q


34/153

Q H Q S H Q

S

G

(P 4, C 4)

P 4 C 4 C 1, C 2, C 3, C 4 C 1C 2, C 2C 3, C 3C 4 C 1C 3, C 2C 4 C 1C 4

N S (C i ) N S (C j ) = N S (C i) N S (C j ) N S (C j ) N S (C i) a1 N S (C i) \ N S (C j ), a 2 C i , a 3 N S (C i ) N S (C j ), a 4 C j , a 5 N S (C j ) \ N S (C i) G[a1, a 2, a 3, a4, a 5] P 5 G

P 5 N S (C 2) N S (C 3) C 1

C 3 N S (C 1) N S (C 3) = N S (C 1) N S (C 2) = C 1C 2

(P 4, C 4)

P 4

(P 5, K 3,z e)

(P 5, K 3,z e) P 5


35/153

f (k) p(n) n f (k)

k p(n) k k

t


36/153

G G 1

w(G)

G w : V (G) R 1 W

W G

W w(G)

(K r e )

K r

r N K r

(G, W ) G K r n 1

G R( W , r )

W G R( W , r ) n R( W , r ) G

W G R( W , r ) n

K r

K r 1 K r e

r N r 2 (K r e)


37/153

(G, W ) G (K r e) n I G

I u v V (G) \ I (N G (u) N G (v)) I I

K 1 K 1,2

n 1 W

|I | < W

V (G) \ I I V (G) \ I

I I

I G |I |

u r 2 K r e u r 2 u

I G (K r e) C K r 2

K r 2 C I K K r e R( W , r 2)

W R( W , r 2)

H I (W 1) + 2 W R( W , r 2) W r

H

W

r I H

L u u I H L1 L C

|C L| < r W L2 L r W C L C |C L | r W


38/153

L1L2 r W 2 W r I L L1 L2 I H I L

W r J G J H = I H J

J I L J L = J \ H J H = I H J J L = J L

J L = I L J L = I L I L J J L x L \ (L1 L2) x

C |C L | r W W J L \ { x} r 2 C C L2 x J L \ { x}

L2 x x (J \ { x}) {x } J I L J

J J L = I L I H I L H

I H I H I H I L

f r (W ) p(n) r f r (W ) W p(n)

W

p(n)

r

W W r

r N G V (G) = X Y G r G (G[X ]) < r (G[Y ]) < r G

r G r

r K r

X k X n n


39/153

X limn log2 X n(n2 ) = 1 1k(X ) E

i,j

i j k(X ) X k X E i,j

i + j = k E i,j i + j = k k X > 1

E i,j limn log2 X nE i,jn = 1 E i,j

max{i + 1 , j + 1 }

E 1,1

r r

r N r r

r

G = ( V, E ) Y (G[Y ]) < r G

Y

|Y \ Y | < R (r, r ) (G[Y ]) < r |Y | = |Y | + 1

G Y Y Y Y := Y G Y

|Y \ Y | < R (r, r ) |Y \ Y | < R (r, r ) (G[Y ]) < r (G[V \ Y ]) < r

G r Y (V \ Y ) G r

G r V = X 0 Y 0 (G[X 0]) < r (G[Y 0]) < r Y G[Y \ Y 0] K r Y \ Y 0

X 0

G[Y \ Y 0]

K r

|Y \ Y 0 | < R (r, r )

|Y 0 \ Y | < R (r, r ) Y = Y 0 |Y 0 \ Y | R(r, r ) |Y 0 | > |Y | Y Y 0

r R(r, r ) G


40/153

r r = 2 r

r r K r r

r N r

(G, W ) G r V (G) =

X Y G[X ] K r

G[Y ] K r

G[Y ] K r G[Y ] I Y w(I Y )

(G[X \ N G (I Y )], W w(I Y )) I X (I Y ) I Y I X (I Y )

r

r N G E (G) =E 0E 1 G (r, G ) G0 = ( V, E 0) rK 2 G1 = ( V, E 1)

G G (r, G ) G (r, G )

r (r,Free (K r )) Free (K r ) K r G = ( V, E ) r

r V = X Y (G[X ]) < r (G[Y ]) < r E 0 E 1 E E 1 = E (G[X ]) E 0 = E \ E 1 G1 = ( V, E 1)

K r G0 = ( V, E 0) rK 2 X

Y G0 rK 2 Y r

r r

(r, G ) G


41/153

(r, G ) (r, G ) (r, G )

r N G

G

(r, G ) (r, G )

(r, G )

(G, W ) G (r, G ) (r, G )

E (G) = E 0 E 1

G

G0 = ( V, E 0)

rK 2

G1 = ( V, E 1) G rK 2 G0

G G1

G0 G1[I 0] I 0 G0

(G, W )

G

H H

H H

K 4 K 4 e K 3 + e K 3 K 1 K 3 + e K 3 K 1

K 4 (K 4 e)


42/153

(K 3 + e) (K 3 + e)

(K 3 + e) (K 3 K 1)

G (K 3K 1) G N G [u] K 3 u V (G)

w(G) = maxuV (G)

{(u) + w(G N G [u])},

(K 3 K 1)

X

G = ( V, E )

U V

x V

U x U x U U V G V \ U U

U 1 < |U | < |V | U |U | < |V | U

U U = V

G G

G

U W

U W G

G


43/153

G

M w(G[M ]) G0

G0

G0

X

X 0

X X 0 X

(G, W ) G X T G

T V G v T Gv G T v T

v T I v

Gv w(I v) I v min{W, w(Gv)} I v W I v

I v w(Gv) v I v = {v} v T v T

Gv v1, v2, . . . , v l v Gv I v = I v1 I v2 . . . I vl

Gv v1, v2, . . . , v l v Gv

I v = I vi w(I vi ) = max {w(I v1 ), w(I v2 ), . . . , w (I vl )} Gv v1, . . . , v l

v Gv U 1, U 2, . . . , U l Gv Gv G0v Gv

U i Gv i w(i) = w(I vi ) G0v X 0 A

(G0v , W ) f (W )lc f (W )n c


44/153


45/153

(C 4,bull) (house,bull )

(C 4,bull) G u, v N G [u] N G [v]

xyz (C 4,bull) G u, v V (C ) N G [u] \ N G [v]

x N G [x]\ (N G [y]N G [z]) V (C ){x } y N G [y]\ N G [z] z N G [z]\ N G [y] G[x,y,z ,x , y ]

G[x,y,z,x , z ] G[x,y,y , x ] G[x,z,z , x ] C 4 xy xz G G[y,z,z , y ] C 4 y z

x N G [y] \ N G [x] G[x,y,x , x ] C 4 x x G[x,y,z,x , x ] x z G

G[x ,y,z ,y , z ] x y G G[x,z,x , y ] C 4

G x

(N G [y]N G [z]) \ N G [x] y (N G [x]N G [z]) \ N G [y] z (N G [x]N G [y]) \ N G [z] {x , y , z } G[x ,z ,y,y , z ]

x y G G[x,y,x , y ] C 4

(C 4,bull)

G u, v N G [u] N G [v] v G

S v u u v S u (C 4,bull) G


46/153

G n3 G G

G G = C 4 C 4


47/153

G G

i(G) G

L(G)2 G


48/153

2s 3s s

n1/ 2 > 0

k k 3

(G, k ) G k


49/153

f (k)nO(1) n G f (k) k

f (k)nO(1)

k G k

k G

k i(G)

k

(C 3, C 4, C 5)

C 4

C 4

(K s , K t,t ) s t (C 3, C 4, C 5)

A x y

(x, y )


50/153

G

X G y y G X

G z X x X z y y

X X

G X G G

G

k C

C

(K s , K t,t )

p M p R(s, t ) R(s, t )

G R(s, t ) G K s G K t G

t N (t, p) N (t, p )

K t,t M p N (t, p )

s t p N (s,t ,p ) N (s,t ,p ) K s

K t,t M p

N (s,t ,p ) = R(s, R (s, N (t, p ))) G (K s , K t,t ) R(s, R (s, N (t, p ))) G


51/153

K s A R(s, N (t, p)) B A G[B ] K s B

N (t, p) A B G[A B ] N (t, p ) G[A B ]

M p

s t k (K s , K t,t )

s t G (K s , K t,t ) n G k

f (k) p(n) f (k) k p(n) n k

M G M N (s,t ,k )

G k N (s,t ,k ) M G

2N (s,t ,k ) O(N (s,t ,k )2kk2) n

M N (s,t ,k ) V M M xy E (G) x V M

y V M M G

v V \ V M V M 22N (s,t,k ) + 2 N (s,t ,k ) k

O((2 2N (s,t,k ) + 2 N (s,t ,k ))2kk2) n

(K s , K t,t )

w(xy) xy


52/153

k G w : E (G) R

k k

G k

s t k (K s , K t,t )

G G x y G

x y z x y a z

w(za ) = max {w(xa ), w(ya)} za a

xa ya

t k k

(K t,t )

A

k A A

A C 4 A C 4

G A G C 4 C 4 = K 2,2 G C 4

G


53/153

A

G = ( U,V,E ) A C 4 G U 0 U 1 V 0 V 1

U 0V 0 U 1V 1 U 0V 1 U 1V 0 P 6

A

G = ( U,V,E )

C 4 H = G[U 0 V 0] C 4 i 1 U i V i U

V i U 0 V 0 U 1 V 1 a U 1 x V 1

a b V 0 c V 0 H C 4

x y U 0 z U 0 a,b,c,x,y,z A

U 0

V 0

G U 1 V 1 U 0 V 0

i > 1 U i V i U 2 a x V 1

x c U 0 U 1 b U 1 y V 0 b

H z V 0 b a,b,c,x,y,z A U 2 V 2

G[U 1 V 0]

P 6 = ( x1, x2, x3, x4, x5, x6) x1, x3, x5 U 1 x2, x4, x6 V 0 a U 0

a, x 1, x2, x3, x4, x6 A G G[U 1V 0] P 6 G[U 0 V 1] P 6

A C 4


54/153

C A C 4 H C H

P 6 H G C U 0, U 1, V 0, V 1

G V (H ) (U 0 V 1) V (H ) (U 1 V 0)

H H P 6 P 6

k C 4

k A

k k 3 c > 1

c


55/153

r r 3

P x P y x cP (x, y) y optP (x)

c 1 y(x) x, cP (x, y(x)) 1c optP (x)

P c c > 1 P

P P 1 + P

P Q P Q (t1, t 2, , ) t1 t2

t1 P Q x P optQ (t1(x)) opt P (x)

x P t2 (t1(x), y ) y t1(x) y x |optP (x) cP (x, t 2(t1(x), y )) | |optQ (t1(x)) cQ (t1(x), y )|

P Q P Q Q P

P Q Q

D G D D G {k} k

D maxdD d 3 D

k k 3

k

k 3 H k = ( V k , E k )


56/153

V k = L1 L2 L3 L4 L5 L6

L1 = {11, . . . , 1k}, L2 = {21, . . . , 2k}, L3 = {31, . . . , 3k(k 1) },L4 = {41, . . . , 4k(k 1) }, L5 = {51, . . . , 5(k 1) 2 }, L6 = {61, . . . , 6(k 1)( k 2) }.

i = 1 , . . . , k 1

S i3 = {3(i 1)k+1 , . . . , 3ik }, S i4 = {4(i 1)k+1 , . . . , 4ik },S i5 = {5(i 1)( k 1)+1 , . . . , 5i(k 1) }, S i6 = {6(i 1)( k 2)+1 , . . . , 6i(k 2) }.

|S i3 | = |S i4 | = k |S i5 | = k 1 |S i6 | = k 2

E k

L1 L2 k (1i , 2i ) E k , i = 1 , . . . , k

(2i , 3(i 1)( k 1)+ j ) E k , i = 1 , . . . , k , j = 1 , . . . , k 1

L3 L4 (3i , 4i) E k , i = 1 , . . . , k (k 1)

i = 1 , . . . , k 1 S i4 S i5 K k,k 1

i = 1 , . . . , k 1 S i3 S i6 K k,k 2

L1 H k

k

i {1, . . . , k 1}, N (S i3) L2 = {2i , 2i+1 }

G = ( V, E ) k S = {v1, . . . , vk} V G S H k H k G L1 S

V L2L3L4L5L6 (GS H k )[V ] = G (GS H k)[S L2L3L4L5L6] = H k G = ( V, E ), G = ( V , E )

G G = ( V V , E E )

k 3 {(3i , 4i ), i = 1 , . . . , k (k 1)} H k

1i

i = 1 , . . . , k H k

M H k (2i , 3(i 1)( k 1)+ j ) i {1, . . . , k } j {1, . . . , k 1} M \ { (2i , 3(i 1)( k 1)+ j )} {(1i , 2i)}

M (u, v), u L2, v L3 i = 1 , . . . , k 1

M i = M [{(1i , 2i ), (1i+1 , 2i+1 )} {(u, v), u S i3, v S i6} {(u, v), u S i4, v S i5}] |M i | 3 (u, v), u S i3, v S i4


57/153

H k k = 3


58/153

M M i M = M \ M i{(u, v), u S i3, v S i4} M (u, v), u S i3, v S i4

(u , v ), u S i3, v S i

6 (u , v ), u S i

4, v S i

5 |M | | M | H k

(u, v), u L3, v L4

G S H k {(3i , 4i), i = 1 , . . . , k (k 1)}

i(G S H k ) = i(G) + k(k 1)

G S G

G S H k G = ( V, E ) D

d G K d,d (D {d})

i(G K d,d ) = i(G) + 1

d D u1d , . . . , u pd p 1

d k 3 k G G1, . . . , G k v V (G) S (v) v G1, . . . , G k |S (v)| = k

T kd (G) = ( G1 . . . Gk) S (u 1d ) H k . . . S (u pd ) H k

i(T kd (G)) = ki(G) + pk(k 1)

k D {d + 1}, d = k T kd (G) (D \ { d} {d + 1}) G K d,d T kd (G)

= 2 , = 1 t1 G GK d,d t2 (G K d,d , M ) (G, M ) M M

K d,d G D i(G) 1


59/153

K d,d M = k(1+2(2( 1)+1)( k 1))

= 1 /k = max( D ) t1 G T kd (G) t2

(T kd (G), M ) (G, M ) M M G T kd (G) G M

G G 2( 1) D

G n2 G i(G) n

2(2( 1)+1) p n

i(T kd (G)) = ki(G) + pk(k 1)

ki(G) + nk (k 1)

k(1 + 2(2( 1) + 1)( k 1)) i(G)

(G, M ) = t2(T kd (G), M ) M T kd (G)

(3i , 4i ) G T kd (G) M |M | = k|M | + pk(k 1)

|M | | M |

i(G) | M | = 1

k(ki (G) + pk(k 1) | M |

1

k(i(T kd (G)) | M |)

D d (D {d}) T d+1d d D, d 2

(D \ { d} {d + 1 }) {2, 3}

> 0

95709569

d = 2 , 3, . . . {3} {4} {k} k 3

D D k 3


60/153

n Qn {0, 1}n

Qn 2n n Qn Qn

n 2 Qn 2n 2

M = {(x1, . . . , x n ) x2 x3 + . . . + xn (2)} M 2n 1 x, y M Qn

x1 Qn [M ] Qn 2n 2

n = 2 n > 2 x1, . . . , x n 2 {0, 1}

(x1, . . . , x n 2, 0, 0), (x1, . . . , x n 2, 0, 1), (x1, . . . , x n 2, 1, 0), (x1, . . . , x n 2, 1, 1).

Qn Qn Qn

2n 2

k


61/153

d (K s , K t,t )

(K s , K t,t )


62/153


63/153

G G S G S


64/153

c > 0 ncn n n

n c1 n

n c2 n n c1, c2 > 0 n

M 1

M 2

M 3 2K 2

M 1 M 2 M 3 M 1 M 2 M 3

M 1 M 2 M 3


65/153

M 4

M 4

M 5 P 4

M 1 M 5

G G

M S 1 M 1 M 1

M S 2

M S 4

M S 1 M S 2 M S 4


67/153

G = ( V, E ) V = S R S G[R] k

R

O(nk

)

R

k 2k

v V R v

v S v R S v S

S

R

v

R R R S R S S R

v R S xv u, v R S u v

R xu xv S

xu xv

G

v

xv = true S R


68/153

M 1

M 5

k, l G A, B G[A]

k G[B ] l n2R (k,l ) 2 n G

O n2R(k,l )+max {k,l } R(k, l) k l

M 1 M 2 M 3

{M1, M 2, M 3} G V 1 V 1

V 2 V 3 V (G)

G V 1

M 4

G G V 1, V 2, V 3 V 3 V 1 V 2

V 2 V 3 G

P G C X G[X ] G M 4 V 1, V 2, V 3 C = V 3 X = V 1 V 2 P


69/153

C, X P V 3 = C X V 1, V 2 V 1, V 2, V 3 M 4 G

C, X P I

xv v X

uv E (G[X ]) (xu xv) (xu xv)

u, v X C (xu xv)

v X v C (xv a) (xv a) a

I G M 4 V 1, V 2, V 3 V 3 = C V 1 V 2 = X

I I {true,false } C j

true (z) (z) z

V 1 = {v | (xv) = false } V 2 = {v | (xv) = true } V 1, V 2, V 3 M 4 G V 1 V 2

V 2 V 3 V 2 V 3

V 1, V 2, V 3 M 4 G V 3 = C I (xv) = false v V 1 (xv) = true

v V 2 a (a) = true I

V 1 V 2 V 3 V 2

V 2 V 3

I

M 4


70/153

G G V 1, V 2, V 3 V 3 V 1 V 2 V 2

V 3 P C, X

P V 3 = C X V 1, V 2 V 1, V 2, V 3 M 4 G

GC G C X G M 4

V 3 = C GC M 4 V 3 V 1 V 2 X G[C ] G[V 1], G[V 2]

GC

[C ] GC

[V 1], GC

[V 2] V 2 V 3 V 3 V 2 G

V 3 V 2 GC

M 2

M 2

M 2

(G, ) G : V (G) 2{1,2,3} (v) v S {1, 2, 3} U S G (v) = S

(G, ) V (G) V 1 V 2 V 3

V 2 V 3

V 3 V 2

{1, 2, 3} v V (v)


71/153

(G, ) G (v) : V (G) 2{1,2,3} (G, )

{1, 2, 3}

u U { } v V (G) \ N (u) (v) (v)

(, ) (2, 3), (3, 2)

u U { } v N (u) U { } w N (u) \ { v} (w) (w) w N (v) \ { u} (w) (w)

u, v G v U {1, } |N (v) U { } | 2

(v) u U { } v, w N (u) U {1, }

(N (v) \ N (w)) U {1, } = (v)

u U { } v, w N (u) U {1, } x U {1, } v, w N (x)

(x) u V (G) 1 (u) U {1, } \ N (u)

(u) u V (G) (u) G N (u) U {1, }

2K 2 P 4 (u)

(G, )

(G, )

U =

(G, )

(G, ) (G , ) (G, ) (G, )

(G , )


72/153

x U {i} i {1, 2, 3} (V 1, V 2, V 3) x V i

{1, 2, 3}

V

u U { }

u, v v V (G, )

u, v u U { } v U { } V 2 V 3 v

u V u v V N (u)N (v) \ { u, v}

u v G

v V v V u U { } v, w N (u)U {1, } z (N (v) \ N (w)) U {1, }

v V u V V V w, z V 1

w, z v V

u U { }, x U {1, } v, w (N (u) \ N (x)) U {1, } u V u V

v, w V 1 V 1 v, w x x V 1 u V (G) 1 (u) v, w U

{1, } \ N (u) v, w

V i i {1, 2, 3} v, w V 1 V u v w u V 1

(u) u V N (u) V 1 u V 2K 2 P 4

N (u) U {1, } u V


73/153

(V 1, V 2, V 3) (G, ) V 2 V 3

G

U {1,2,3} = U {2,3} = U {1,2} = U {1,3} =

U {1,3} =

x U {1,2} vx

z U {3} x, y N (z) U {1,2} (vx vy)

x, y U {1,2} xy E (G) (vx vy) (vx vy)

(G, )

(V 1, V 2, V 3) (G, )

V 1 = U {1} {x | (vx ) = false } V 2 = U {2} U {1,2} \ V 1 V 3 = U {3}

U {1,2,3} = U {2,3} = U {1,2} U {1,3} G

(G, )

V 2 = U {2} V 3 = U {3} U {1,3} \ V 1

V 1 = U {1} U {1,2} uU { 2 }

|N (u) U { 1 , 3 } | 2

N (u) U {1,3}

(G, ) U {1,3} V 2 V 3

V 1 V 3 V 2 V 3 V 3 V 2 U {1,2}, U {1,3}

V 1 U {1,2} V 1 U {1,3}


74/153

u U {2} v, w N (u) U {1,3} U {1,2} v w

v, w U {1,2}

U {1,2} V 1 U {1,3} V 1

U {1,2,3} = U {2,3} =

U {1,2,3} = U {2,3} = U {1,2} = U {1,3} = u U {1,2} w N (u) U {1,3} (w) = {3} (v) v N (u) 2 G U {1,3}

w w U {1,3} u U {1,3}

(w ) = {2} w N (u ) U {1,2} (v) v N (u ) U {1,2} G U {1,2}

w w w w O(n2)

O(n2) O(n2) (G, )

(G, ) O(n2)

O(n2) (G, )

V 1, V 2, V 3 (G, ) H =G[U {1,3}] H G U {1,3} u U {1,2}

u V 1 u V 3

H w

u V (H ) = U {1,3} H w


75/153

v V 1 v V 3 (v) = {1, 3} v

H z H z v H G z u z V 1 V 3

V 1, V 2, V 3 (G, ) (z) = {1, 3} z u

w H w x xw E (G)

ux E (G) V 1, V 2, V 3 x ,z ,w,v u

(G, ) H w x v z

V 1, V 2, V 3 O(n2)

u V 3 V (H ) H

w u V (H ) H

u V 3 V (H ) V 1, V 2, V 3

u V 2

u

V 3 V (H )

w u

H w v V 3 u V 3 V 1, V 2, V 3

u V 3 V (H ) w u

H V 1, V 2, V 3

N (u) V (H ) V 3 = w (G, )

w w w

O(n2) (G+ , + ) (G, ) u


76/153

w u w

U +

{1,2} U

+

{1,3} G

U +

{1,3}

v, v + v, v U {1,3} H G[U {1,3}]

u v, v u v v (u)

(G, ) (u) = {1, 2} u v

w (G+ , + ) u H

v v (v)

+ u v

+ (v) = {1, 3} w H v

w H v v = w (G+ , + ) (v)

(G+ , + ) (w) = {3} + (v) = {1, 3}

U +

{1,3} U +

{1,2} u w u w

M 2

(G, ) (G, )

(V 1, V 2, V 3) (G, ) u V 2 v V 3 uv E (G) u, v

O(n2) u, v (u) = {2} (v) = {3}

u, v (G+ , + )


77/153

U +

{1,2,3} = U +

{2,3} = (G+ , + )

M S 1 M S 2

(X X Y ) (X X Y ) X Y

X X

(X X Y ) (uvX ) (uvX ) (wzY ) (wzY ) u ,v,w,z

(X X Y ) (u v Y ) (u vY ) u, v X

(u v X ) (u w X ) (v w z) (v w z) (v w z) u ,v,w,z

M S 4


78/153

G G V 1, V 2, V 3 V 3 V 1 V 2

V 2 V 3

I m C 1, . . . , C m v1, . . . , vn

J i j vi C j J i j vi C j

I G I y1, . . . , ym vi

vi C j xi,j xi,j y j x i,j xi,j xi,j xi, i {1, . . . , n }

j J i J i G I M S 3 I

I {v1, . . . , vn } {true, false } C j

C j true (vi ) (vi) V (G I )

V 1 = x i,j j J i (vi ) = false x i,j j J i (vi ) = true

V 2 = x i,j j J i (vi ) = true x i,j j J i (vi ) = false

V 3 = y j j {1, . . . , m }

V 1 V 2 G I V 3 V 2 V 3

y j V 3 xi,j xi,j V 2 vi vi C j true G I M S 3

G I M S 3 V (G I ) V 1, V 2, V 3 V 1, V 2

V 3 V 2 V 3 V 3 = {y j | j {1, . . . , m }}

y j V 1 V 2 V 3 y j V 3 xi,j xi,j

{y1, . . . , ym } V 3 y j V 2 j


79/153

y j V 3 y j V 1 j z {y1, . . . , ym } y j z xi,j x i,j i z V 3 V 3

x i,j xi,j z V 1 V 1 z V 2 z {y1, . . . , ym } y j z

V 3 V 3 = {y1, . . . , ym } : {v1, . . . , vn } {true,false }

i {1, . . . , n } (vi ) = true xi,j V 2 j (vi ) = false I

V 1 = x i,j j J i (vi ) = false x i,j j J i (vi ) = true }

V 2 = x i,j j J i (vi ) = true x i,j j J i (vi ) = false

i {1, . . . , n } xi,j xi, j J i J i (vi) = true xi,j V 2 j

x i, V 1 J i V 2 xi,j V 2 j J i V 1 (vi ) = false xi,j V 1 j J i

x i, V 2 J i C j y j V 3

x i,j

xi,j

V 2

vi vi

C j

true xi ,j xi ,j y j V 1

vi vi false C j

M S 4

G G V 1, V 2, V 3 V 3 V 1

V 2 V 2 V 3


80/153

I m C 1, . . . , C m v1, . . . , vn


I G I G+ I G I

{y1, . . . , ym } i {1, . . . , m } yi z {y1, . . . , ym } G+ I yi z G I

G+ I M S 4 I

G I M S 4 V 1, V 2, V 3 V 3 = {y1, . . . , ym } M S 4 G

+ I I G

+ I

M S 4 G+ I M S 4 V 1 V 2

V 3 V (G I ) V 1, V 2 V 3 V 2 V 3

V 3 = {y1, . . . , ym } G+ I V 1, V 2, V 3 M S 3 G I

I y j vi

vi vi C j vi C j 1 C j 2 vi C j 3 C j 4

G+ I xi,j 1 , x i,j 2 , x i,j 3 , x i,j 4 y j y j V 1 V 1

x i,j 1 , x i,j 2 , x i,j 3 , x i,j 4 V 2 V 3 G+ I [V 2 V 3] y j V 1 y j V 2

V 3 {y1, . . . , ym } V 3 xi,j xi,j {y1, . . . , ym } V 3 V 3 = {y1, . . . , ym }

M S 2

G V 1, V 2, V 3 V 2


81/153

V 3

I m C 1, . . . , C m v1, . . . , vn


I G+ I G I G

+ I yiy j , i , j {1, . . . , m }

{y1, . . . , ym }

G I M S 2 I

G I M S 4 V 1, V 2, V 3 V 3 = {y1, . . . , ym } M S 2 G I I G I

M S 2 G I M S 2

V (G I ) V 1, V 2, V 3 V 2 V 3

a,b,c V 2V 3 V 2 V 3

a, b V 2 c V 3 c V 2 V 2 V 3

y j V 2 j {1, . . . , m } y j V 3 j {1, . . . , m } yi V 1

yi V 1 xi1 ,j 1 , x i2 ,j 2 , x i3 ,j 3 x i1 ,j 4 i1, i2, i3 C j

j1, j 2, j 3, j 4 = j y j x i1 ,j 1 , x i2 ,j 2 , x i3 ,j 3 , x i1 ,j 4 V 2 V 3 xi1 ,j 1 , x i2 ,j 2 , x i3 ,j 3

x i2 ,j 2 , x i3 ,j 3 , x i1 ,j 4 V i (i {2, 3}) xi2 ,j 2 xi1 ,j 4

V 2 V 3 {y1, . . . , ym } V 3

x i,j xi,j I y j V 3


82/153


83/153

Y

2C 4 C 4 C 4 C 4 2C 4

C 4

2C 4 C 4 C 4

2C 4 C 4 C 4

G

Y

C 1

C 2

C 4 G C 1 C 2 G

G G

AB G G A = A1 A2 B = B1 B2 A1 B1

A2 B2 C 4 G

G

A1 B1 A2 B2

A2 B2

A1 B2 A2

B1 A2 B2

A1 B2

B1

A2

C 1 C 2

C 1 C 2 A2 B2

Y (2C 4, C 4 C 4)


84/153

n Y

n (2C 4, C 4 C 4) (2(n log

2 n ) ) (2C 4, C 4 C 4)

= M 1 M 3


85/153

G G

M M

M S 1

d d 3


86/153

S k (C 3, . . . , C k , H 1, . . . , H k ) 3 C k k H k

G (G) k G S k G S k (G)

G S k (G) M (M ) = sup {(G) : G M }

M X M (M ) <

X

12

i 1i

12 j 1

j

1 2k 1

k

i

S i,j,k H i

P = N P

M M (M ) =

M

H k

G (G) =


87/153

H k , H k+1 , . . .

claw P 7

(G) = G S k G S k

k G S k G S i,j,k i , j ,k 0

S i,j,k S 1,1,1

S 1,1,2 S 1,1,3

S 1,2,2 E S 1,2,2

E

|B | |W | S 1,1,3

F F


88/153

G W B W G G[B ] G

G[B ] W B

G w : E (G) R B, W V (G)

B , W B B W W eM w(e)

M = {xy E (G) : x, y B }

B = W =

B = W = w

(G,B,W ) (G,B,W ) = 2 |V (G)| | B | | W |

(G,B,W ) 0 |V (G)| = |B | = |W |


89/153


90/153

V |(G)| = 2 (G, V (G), W ) G

G

x B, N (x) = 1 (G, B N (x), W ) x

x, y B z V (G) xz,yz E (G) (G,B,W {z}) z

G w,x,y,z V (G) wx,wy,wz,xz,yz E (G) xy E (G) w z

x y (G, B {w, z }, W {x, y})

x1x2, x2x3, x3x4, x4x1 E (G) x1x3, x2x4 G x1 B (G, B {x3}, W {x2, x4}) C 4

C 4

G K 4 (G, V (G), V (G))

G x, y1, y2, y3, y4 V (G) xy1, xy2, xy3, xy4, y1y2, y3y4 E (G) y1y3, y1y4, y2y3, y2y4 E (G)

x y1, y2, y3, y4 (G, B {y1, y2, y3, y4}, W {x})

G w,x,y,z V (G) wx,xy,xz,yz E (G) wy,wz E (G)

w x x B (G,B,W {w}) w B (G,B,W {x})

G (butterfly, diamond, K 4)


91/153

K 4

K 4

w

x

|B | |W | (k, l)

(k, ) G B, W V (G)

k l B , W

B B W W |B | k |W |

G G k

(k, 0)

(k, )

(G,B,W ) w


92/153

k

|B | > k |W | >

(B, W ) V (G) (B, W )

G P 3 K 3

p(n)3max {k, } p(n) k

S 1,1,3

S 1,1,3

(C 5, C 6, . . . )


93/153

(C 5, C 6, . . . )

(G,B,W ) G (C 5, C 6, . . . ) x B

x x W

x G

x B x W G

G G G

(G,B,W ) (C 5, C 6, . . . )

S

G

G

S S

W B G

G S 1,1,3 W B G

G W B G

G W = B B B

B B B B


94/153

x B B x G (butterfly, diamond, K 4) N (x) G[N (x)]

G[N (x)] N (x)

x B |N (x)| 3 V (G) = N (x) {x} G[N (x)] y1, y2, y3

x z N (x) x z y1

z B z B z z y1, y2, y3 z y2 y3

G[x, y2, y3, y1, z , z ] S 1,1,3 |N (x)| 3 G

B

x, y B B

G

C v1, . . . , vk C x V (G)

d C d C

x V (G) \ V (C )

x C G (diamond, butterfly, K 4)

k 6 x v2 G[v2,x ,v 1, v3, v4, v5] S 1,1,3

k 7 x x C k 7 x x

C x v1 vi 3 i k2 +1 i = 3 G[v3, v2,x ,v 4, v5, v6] S 1,1,3 i = 4 G[v1, v2, vk ,x ,v 4, v5]


95/153

S 1,1,3 i 5 G[v1,x ,v k , v2, v3, v4] S 1,1,3

k 7 x 3 x C x C v1, vi , v j

C i < j i > 3 G[v1, v2, vk ,x ,v i , vi+1 ] S 1,1,3 i = 3 x v5

G[v1, v2, vk ,x ,v 5, v4] S 1,1,3

k 8 3 C k 8 x 3 x

x v1, v2, vi 6 i k 1 G[v

i, v

i 1, v

i+1,x ,v

2, v

3] S

1,1,3

k = 7 x 3 x x C x

x 4 x v1, v2, v4, v6 v3, v5, v7 G[v4, v3, v5,x ,v 1, v7]

S 1,1,3 x x x

v1, v2, vi i = 5 v5 G[x, v1, v2, v5] x v1 v2 v3 v7

v4 v6 v5 v5

G (G,B,W {y}) y x

7

G

C

k 7

vi k

x V (G) \ V (C ) C x C C

(butterfly, diamond, K 4) C


96/153

C C C

x y C x y C i, j i < j x

vi vi+1 y v j v j +1 vi+2 , . . . , v j 1 C k

G[vi , vi+1 ,x ,v i+2 ] vi+1 vi+2 v j 1 v j G (butterfly, diamond, K 4)

j i + 2

x1x2x3x4 G x2 x3 G

x1 x4

j i 0 mod 3 vi+2 vi+1 vi+3

v j 1 v j vi+2

j i 1 mod 3 vi+1 vi+2 vi+3 v j 1 v j

vi+1

C x, y j i = 2 mod 3 (G,B,W {vi+2 }) j i 0 mod 3 (G,B,W {vi+1 })

j i 1 mod 3

j i 2 mod 3 x, y x1, . . . , x t C xt+1 := x1 vi i vi i +1

x i ii ii < i i+1 < = ii + k x i , x i+1 C xi

vi i

vi i +1

xi+1

vi i +1

vi i +1 +1

vi i +2

, . . . , vi i +1 1

C {vi i +2 , . . . , v i i +1 1}

x i xi+1 i G[x i , vi i , vi i +1 ] vi i , vi i +1

vi i +1 vi i +2 ii+1 = ii + 2 vi i +1 +1 vi i +3 vi i +4


97/153

vi i +1 vi i +1 +1 vi j vi j +1 j vi i +1

vii

vij

vi j +1 j N (C )

C C N (C ) B

(G, V (G), V (G))

C C white black black

C

C C (G, V (G), V (G))

G 7

G C

v1, . . . , v6 x

4 x

x v1, v3, v5 v2, v4, v6

x x

v1 v3 G[v1,x ,v 2, v6, v5, v4] S 1,1,3 x

x x x

v1, v2 v4 x G[x, v1, v2, v4] v4 v3 v5 v2 v1 v6

x v1, v2 v6 v6 v6


98/153

C 6 S 1,1,3

C = v1 v2 v3 v4 v5 v6 v1 x C v1, v2, v4 (G,B,W {v6})

x y x v1, v3, v5 y

v1 v2 x y G[v1, v6,y,x ,v 3, v4] S 1,1,3 G[v1, y,v2, x]

x v1, v3 v5 G[v1, v2, v3, x] G[v3, v4, v5, x] G[v5, v6, v1, x] C 4 v1, v3, v5 v2, v4, v6, x

C = v1 v2 v3 v4 v5 v6 v1 G x C v1, v3, v5

y1, y2, y3 yi x i xi+3 (G, V (G), V (G))

C G x y (G,B,W {x})


99/153

z, z x z z x z G[v1, v2, v6,x ,z ,z ] S 1,1,3

3b

v1

C G x C

(G, V (G), V (G))

y x x v1 v2 y

v1 v4 x y G[v1,x ,v 2, y] G[v1, v6,x ,y,v 4, v3] S 1,1,3

G diamond, butterfly, K 4 a1 v1 v2 a2 v3 v4 a3 v5 v6

a1 v1 v2 v3 v6 a2, a 3, v4, v5

a i

C G a1, a2, a 3 v1&v2 v3&v4 v5&v6 (G, B{a1, a 2, a 3, v1, v3, v5}, W

{v2, v4, v6}) (G, B {a1, a2, a3, v2, v4, v6}, W {v1, v3, v5})

(G, V (G), V (G))

a1 v1 v2 a2 v3 v4 G[v2, a1, v1, v3] v2 v3

v2 a1, v1 v3 v6 v4 v5 G[a2, v3, v4, v5]


100/153

C G (G, V (G), V (G))

a1 v1 v2 a2 v4 v5 v3 v6 v3

G[v2, a1, v1, v3] v2 v6 v1 v1 v2 v3 v6

C G a1, a 2 v1&v2 v4&v5 C

(G, B {v1, v2, v3, v4}, W {v3, v6, a 1, a 2}) x v3 v6

x x a1 C

C G a1, a 2 v1&v2 v4&v5 x

a1 (G, B {v1, v2, v3, v4}, W {v3, v6, a1, a2})

v3 v6 x a1 v1 v2 a1

y y x G[v6,x ,v 5, v1, a 1, y] S 1,1,3 z x z

z x G[v6, v1, v5,x ,z ,z ] S 1,1,3

C G a1, a 2 v1&v2 v4&v5

G

a1 v1 v2


101/153

v3 v6 v1 a1, v2 v6 v3 v4 v5 v4

v5 v1

v2

C G a1 v1&v2 (G, B {v1, v2, v4, v5}, W {a1, v3, v6})

x z, z

x,z,z x v1 v4 G[v1, v2, v6,x ,z ,z ] S 1,1,3

C G

6

C

C 5 S 1,1,3


102/153

x v1, v2 v4 x

G[x, v1, v2, v4] v4 v3 G[v2, v1,x ,v 3] v2 v1

x

C G x (G,B,W {x})

a1, a2 G (diamond, butterfly, K 4) a1 a2

a1

v1

v2

a2 v3 v4 v5 G[v1, a1, v2, v5] v1 a1 v2 a2 v3

v2 v5

C G x y C (G,B,W {y})

x, x z G[z,x,x ] P 3 v1 z G[v1, v2, v5, z ,x ,x ]

S 1,1,3

vi S 1,1,3

C G

x


103/153

x

z, z G[x,z,z ] P 3 x v1 v2

x v1 v2 v3 v5 v4

y v4 y v2 G[v2, v3, v4, y] C 4 y

y x y G[v4, y,v5, v3, v2, x] S 1,1,3

x z z y y v4 G[v4, v3, v5,y,x ,z ] S 1,1,3

x v1 v2 x

C G x (G,B,W (N (x) \ C )

5

S 1,1,3

t

(S 1,1,3 + tK 2)

(G,B,W ) t G


104/153

t M 2t (G, B M, W )

n2t M

M M tK 2 S 1,1,3

F

F

F S i,j,k i , j ,k 0 F

F

F S i,j,k F F = S 1,2,2

P 7 P 6 P 5 + K 1 P 4 + P 2 2P 3 F

S 1,13 + 6 K 2

S 1,1,3

S 1,2,3 S 1,1,3 S 1,2,2 S 1,2,3

S 1,2,3

P 7 P 8


105/153

P 2 + 4 K 1

P 3 + 3 K 1

2P 2 + 2 K 1

P 3 + P 2 + K 1

K 1,3 + 2 K 1

P 4 + 2 K 1

3P 2

S 1,1,2 + K 1

P 5 + K 1 P 7

K 1,3 + P 2

P 4 + P 2 P 7

2P 3 P 7

S 1,2,2

S 1,1,3

P 6 P 7

F F


106/153

(C k , C k+1 , C k+2 , . . . ) k > 5 k = 5


107/153


108/153

G

G (G) G k G k k k

k k 3

4 8

3

k k


109/153

k k

K 2,3

k k 3


110/153

S 1,2,2

H

K 1,5

P 4 + P 2

2P 3

F (K 3, F )

F F F = K 1,5 Free (K 3, F ) F

S 1,2,2

,H,cross,P 4 + P

2, 2P

3 P

6

Free (K 3, F )

F k + 1

Free (C 3, C 4, . . . , C k) k (K 3, K 1,5)

Free (K 3, H ) Free (K 3 S 1,2,2) Free (K 3, cross )

Free (K 3, P 4 + P 2) Free (K 3, 2P 3)


111/153

(K 3, P 6)

F Free (K 3, F )

F Free (K 3, F )

F H S 1,2,2 cross P 6 Free (K 3, F )

Free (K 3, H ) Free (K 3, S 1,2,2) Free (K 3, cross ) Free (K 3, P 6)

(K 3, F ) F

Free (K 3, F ) F F

Free (K 3, F ) F v F 0 F 0 = F v

F 6 G Free (K 3, F )

F 0 = P 3 + P 2 F 0 H S 1,2,2 cross (K 3, F 0)

(K 3, F )


112/153


113/153

P 2 + 4 K 1

P 3 + 3 K 1

2P 2 + 2 K 1

P 3 + P 2 + K 1

K 1,3 + 2 K 1

P 4 + 2 K 1

S 1,1,2 + K 1

K 1,4 + K 1

P 5 + K 1

F K 3, F


114/153

Free (K 3, S 1,1,3) Free (K 3, K 1,3 + K 2)

Free (K 3, S 1,1,3) Free (K 3, K 1,3 + K 2)

4

S 1,1,3 (K 1,3 + K 2)

k X X [k] X k

X X [k]

G G

G

k X X (k) X k

X (k) X

X X

Free (K 3, S 1,1,3)

Free (K 3, K 1,3 + K 2)

G (K 3, S 1,1,3) C G = C

C = v1 v2 v2k v2k+1 v1 G 2k + 1 k 3 v V (G) \ V (C )


115/153

C v v1 v v4 G K 3

v v2k+1 , v2, v3, v5 G[v4, v3, v5, v,v1, v2k+1 ] S 1,1,3 v v4 v v3 G[v1, v,v2k+1 , v2, v3, v4] S 1,1,3

v3 v1 v v5 G[v1, v2, v2k+1 , v,v5, v4] S 1,1,3

G = C

G (K 3, K 1,3 + K 2) C 2k+1 k 3 k 4 G = C 2k+1 k = 3 |V (G)| 28

C = v1 v2 v2k v2k+1 v1 2k +1 G k 4 v V (G) \ V (C )

C v1 G K 3 v v2k+1 , v2 {vi , vi+1 }

i = 4 , 5, . . . , 2k 2 v vi , vi+1 G K 3 v {vi , vi+1 } v {vi , vi+1 }

G[v1, v2, v,v2k+1 , vi , vi+1 ] K 1,3 + K 2 v v4 v {v4, v6, . . . , v2k 2}

{v5, v7, . . . , v2k 1} G[v2k 2, v,v2k 3, v2k 1, v2, v3] K 1,3 + K 2 v v5

v {v5, v7, . . . , v2k 1} {v4, v6, . . . , v2k 2} v v2k G K 3 G[v5, v4, v6, v,v2k , v2k+1 ]

K 1,3 + K 2 G = C k = 3 v V (G) \ V (C ) v1

v {v4, v5} v v4 v {v2, v3, v5, v7} v

v6 G[v6, v5, v7, v,v2, v3] K 1,3+ K 2 v V (G) \ V (C )

vi V (C )

{vi , vi+3 }

V (C ) \ {vi , vi+3 }

{vi , vi+4 } V (C ) \ { vi , vi+4 } 7

U j j

| U 1 | 7 |U 1| > 7 z, z U 1 {vi , vi+3 } V (C ) \ { vi , vi+3 } i


116/153

G K 3 z, z G[vi , z , z , vi+1 , vi+4 , vi+5 ] K 1,3 + K 2

U 1 U 2 x U 1 y, z U 2 x

{vi , vi+3 } V (C ) \ {vi , vi+3 } G K 3 y, z G[x,y,z ,v i , vi+4 , vi+5 ]

K 1,3 + K 2

U 2 U 3

i 4 U i U 4 = u4, u3, u2, u1 U 4 C u j U j u1

vi G[vi , vi 1, vi+1 , u1, u3, u4] K 1,3 + K 2

V (G) = V (C ) U 1 U 2 U 3 |U 3 | | U 2 | |U 1 | 7 = |V (C )| |V (G)| 28

C 5

G (K 3, S 1,1,3) C 5 G

G (K 3, S 1,1,3) C = v1 v2 v3 v4 v5 v1 G G = C G

v V (G) \ V (C ) G K 3 v C v

C V (G) \ V (C ) i C N i i {0, 1, 2}

i = 1 , . . . , 5 V i N 2 vi 1, vi+1 V (C ) i V i

V j vi v j C V i |V i | 2

V i G K 3


117/153

N 0 xy x, y N 0 y z N 1N 2

vi V (C ) z G K 3 z x, v i 1, vi+1 G[vi , vi 1, vi+1 , z ,y,x ] S 1,1,3

x N 1 N 2 N 0 x N 1 N 2 z, z N 0 vi V (C ) x G K 3

x vi 1, vi+1 x vi 2, vi+2 x vi 2

G[x,z,z , vi , vi 1, vi 2] S 1,1,3

|N 1 | 5 x, x N 1 vi V (C ) G[vi ,x ,x , vi+1 , vi+2 , vi+3 ] S 1,1,3

V i V j V i V j G K 3

V i V j x V i V j x V i y, y V j

j = i + 1 G[vi 3, y,y , vi 2, vi 1, x] S 1,1,3

V i V j N 0 V i V j i = 1 j = 4

x N 0 y V 1 x z V 4 G[v3, v4, z , v2, y,x ] S 1,1,3

x V 4 G[x,z,z ,y,v2, v1] z, z V 4 S 1,1,3

G N 0 N 0 N 1N

2 V 0

N 0

V i G0 G V 0 G

G0 G G0 G G0

G0


118/153


119/153

G (K 3, K 1,3 +K 2) C = v1 v2 v3 v4 v5 v1

G G = C G v V (G) \ V (C ) G K 3 v

C v C C V (G) \ V (C )

i C N i i {0, 1, 2} i = 1 , . . . , 5 V i N 2 vi 1, vi+1 V (C )

i V i V j vi v j C V i |V i | 7

V i G K 3

|N 1 | 10 x, x , x N 1 vi V (C ) G[vi ,x ,x , x , vi+2 , vi+3 ] K 1,3 + K 2

x, x , x G K 3

V i V j V i V j G K 3

V i

V j

V i

V j j = i + 1 x V i y, y , y V j G[vi+2 , y,y , y , vi 1, x]

K 1,3 + K 2

w N 0 V i w N 0 z, z , z V i

G[w,z,z , z , vi+2 , vi+3 ] K 1,3 + K 2

N 0 N

0 w N

0

z, z , z N 0 G[w,z,z , z , v1, v2] K 1,3 + K 2 G K 3

V i w, w N 0 V i

z, z , z V i {w, w } G[vi 1, z , z , z ,w,w ] K 1,3 + K 2


120/153

V i V j N 0 V i V j i = 1 j = 4

w N 0 y V 1 V 4 w V 4

w z, z V 4 G[v3, v4, z , z ,w,y ] K 1,3 + K 2

x N 1 N 2 N 0 x N 1 N 2 C x

x x N 0 K 1,3 + K 2

G[N 0] V i G

V i N 0 G N 0

N 1N 2 V 0 N 0 V i G0 G V 0

V (G) \ V (G0) G G0

G0

V i V j V 0 = V i 1 V i+1

x V i V i 1 V i+1 y V i+1 x V i 1 x

z, z V i 1 G[x,z,z , vi 1, vi+2 , y] K 1,3 + K 2 G0

G0

G V 1 V 3 V 4

V 2 V 5


121/153

V 2 V 5 G0 (V 1, V 2 V 5 V 0) G0

V 2 V 5 G0 (V 1, V 0) G0

V 2 V 5 G0 V 0 V 1 V 2 w V 0 x V 1 y V 2

w x V 2 y V 1 x V 1 w

z V 2 x V 1 V

1 x

1, x

2, x

3 z w

G0[z, x 1, x2, x3,x ,w ] K 1,3 + K 2

G0 V 1 V 2 G0

G0 C G0 C

C C x

x V 1 V 2 x V 1 G0 x V 0

V 2 x y V 2 w V 0 G0 G0 y

x y, w w V (C ) (V 1V 2)

|V (C ) (V 1 V 2)| 4 V 1 V 2 V 0 |V (C ) | 12 C

x V 0 C V 1 V 2 V 0 x V 1 V 2 z, z , z

x z, z V 1 z V 2 G K 3 C z

z z z V 2


122/153

(K 3, S 1,1,3) (K 3, K 1,3+ K 2)

(K 3, S 1,2,3, S 1,1,2 + P 2)

(K 3 S 1,2,3 S 1,1,2 + P 2) S 1,2,3 S 1,1,2 + P 2

P 4+ P 2 (K 3, P 4 + P 2)

S 1 , 2 , 3 S 1 ,1 ,2 + P 2

S 1,2,3 S 1,1,2 + P 2

G (K 3, S 1,2,3, S 1,1,2 + P 2) G 4 G

G ab G K 3 G[N (a) N (b)] X = V (G) \ (N (a) N (b))

G[X ] G 4 G[X ] K 3

v1 v2k+1 v1 k 2 w1, w2, . . . , w q a wq = a

w1 = v

i i {1, . . . , 2k + 1} q = 3 w

2 N (a) \ { b}

w4 = b w2 vi 1 vi+1 G K 3 w2

vi+2 G[vi , vi 1, vi+1 , vi+2 , w2, w3, w4] S 1,2,3 vi k

w2 G K 3


123/153

(K 3, S 1,2,3, S 1,1,2 + P 2)

G (K 3, S 1,2,3, S 1,1,2 + P 2) C G = C

C = v1 v2 v2k+1 G x vi C vi 1 vi+1 K 3 vi 2

G vi , vi+1 vi 1, vi 2 x, vi+3 , vi+4 S 1,2,3 x {vi+3 , vi+4 } S 1,1,2 + P 2 x {vi+3 , vi+4 }

x vi 2 vi k G

K 3

G (K 3, S 1,2,3, S 1,1,2+ P 2) C C

G C = v1 v2 v3 v4 v5 v6 v7 v1 C x y y

x y v1 x v2 v7 G K 3 x

v4

v5

G[v1, v2, v7,x ,y,v 4, v5]

S 1,1,2 + P 2

x v4 G K 3 x v3 v5 x v6 G[v1,x ,v 2, v3, v7, v6, v5]

S 1,2,3 G[v6, v5, v7,x ,y,v 2, v3] S 1,1,2 + P 2 x y C

B G B

B

G B |V (B )| 3 x B x

B B B G

G


124/153

G (K 3, S 1,2,3, S 1,1,2+ P 2) C k 8 x x

C

x vi vi 1 vi+1 K 3 vi 2

G vi , vi+1 vi 1, vi 2 x, vi+3 , vi+4 S 1,2,3 x {vi+3 , vi+4 } S 1,1,2 + P 2 x

{vi+3 , vi+4 } x vi 2 vi x vi

G

u v u v v u

u, v G

G (K 3, S 1,2,3, S 1,1,2 + P 2) P

G

8

x

2 P x P

P v1 v2 vk k 8 x vi 2 < i k 1 x vi 2

x vi+2 x vi 1 vi+1 G K 3

i < k 3 G[vi ,x ,v i+1 , vi 1, vi 2, vi+3 , vi+4 ] S 1,2,3 x {vi+3 , vi+4 } S 1,1,2 + P 2 x {vi+3 , vi+4 }

i k 3 k 9 k = 8 , i k 2 G[vi ,x ,v i+1 , vi 1, vi 2, vi 4, vi 5]

S 1,2,3 x {vi 5, vi 4} S 1,1,2 + P 2 x {vi 5, vi 4} k 9 k = 8 , i = k 3

x vi 2 k = 8 i = k 3 = 5

k = 8 , i = 3 x v7 G


125/153

v6 y x y v8 v4 v2 x

v2 v4 G K 3 x v1 G[y, v6, v8, v4, v3, v1, x] S 1,1,2 + P 2

x v1 G[y, v4, v2, v1, v6, v7, x] S 1,2,3

G

P G x V (G) \ V (P ) P

G (K 3, S 1,2,3, S 1,1,2 + P 2, C 7, C 8, P 8)

G G P 6 3 3 G

Q C 6

Q

G Q Q a,b,c,d,e,f,g V (G) a b c d e f a

g e G {a,b,c} 5

V a a b c

V b V c V a


126/153

V ac a c b

W {a,b,c}

V a , V b, V c V ac G K 3 W

3 G

uv G[W \ {e, g}] u, v {e, g} G[e,d,g,f,a,u,v ] S 1,1,2 + P 2

u, v d, f u f G[f,u,e,g,a,b,c ]

S 1,2,3

W W 0 W 1 G[W 1] G[W ] e g W 0 = W \ W 1

uv G[W 1] u, v {d, f }

e

ug

G[W 1]

u = e g d, f u d, f G[e,f,g,u,d,c,b ] S 1,2,3 u

d, f uv G[W 1] u, v = e, g G (K 3, C 7) u, v {d, f }

u, v d, f u, v g u, v e

u e G[e,g,u,v,f ,a ,b ] S 1,2,3

G[W 1] u W 1 \ { e, g} e, g e g

G K 3 u W 1 \ { e, g} u e, g u f d G[f,u,e,g,a,b,c ] G[d,u,e,g,c,b,a ]

S 1,2,3 v u W 1 v d, f v


127/153

f G[f,e,v,u,a,b,c ] S 1,2,3 u W 1 \ {e, g} e, g

W 1(g) W 1 e W 1(e) g e W 1(e) g W 1(g)

W 1(e) W 1(g) w W 1(g) w W 1(e) g d, f w

d, f w f G[f, w ,e,w,a,b,c ] S 1,2,3

e d, f W 1(g) {d, f } W 1(e) d, f

v V a V c v = d, f ww G[W 1] w, w v v V c w W 1(e) w W 1(g) G[c,v,b,a,d,w,w ] S 1,2,3 dw E (G) S 1,1,2 + P 2 dw E (G)

u, v W 1(e) uf,vd E (G) ud,vf E (G) u, v = e

G[d,v,c,b,e,f,u ] S 1,2,3

d f W 1(e) f W 1(e)

V a V a = V 1a V 2a V 1a W 1(e) W 1(g) V 2a W 1(g) W 1(e)

V c V c = V 1c V 2c V 1c W 1(e) W 1(g) V 2c W 1(g)

W 1(e) G K 3 V 1a V 1c V 2a V 2c

W 0

V a V c

u W 0

u

v V a V c ww G[W 1] w, w v

w v G[v,u,w,w ,a,b,c] S 1,2,3


128/153

W 1(g) W 0 V ac w W 1(g) u W 0 v V ac G K 3 e

v G[v,u,a,b,w,e,d ] S 1,2,3

X V ac W 1(g) Y V ac X W 1(e) G

K 3 G K 3 V 2a V 2c W 1(e)W 0{b}X V 1a V 1c W 1(g)Y

V b {a, c} G 3

G Q

a,b,c,d,e,f

P 6

{ab,bc,cd,de,ef }

G

P 6 {b,c,d,e} 8

V b b c,d,e

V c, V d , V e V b

V bd b d c e

V ce V be V bd

W {b,c,d,e}

V b V e a V b f V e af E (G) a u V e \ { f } G[a,b,c,d,e,u,f ]

Q a V e u V b \ { a}, v V e G[b,c,d,e,v,u,a ]

Q V b V e

W V b V e V b V e w W u V b v V b G[b,v,u,w,c,d,e ]

S 1,2,3 V b V e


129/153

W V b V e w W V b V e G[a,b,c,d,e,f,w ] C 7

W W b, W e , W 0 W b V b V e W e V e

V b W 0 V b V e W b W e

W b, W e W b W e u W b v W e G[u,a,b,c,d,e,f,v ] C 8 P 8

W e = W V e |W b| 1 u, v W b G[a,u,v,b,c,e,f ] S 1,1,2 + P 2

W W uv u W b G K 3 v a G[v,u,a,b,c,d,e,f ]

P 8 u v V b u, v W 0

P P 6 = {ab,bc,cd,de,ef } W b = P 7 = {ya,ab,bc,cd,de,ef } W b = {y} P

1, 2, . . . , 6 1, 2, . . . , 7 k P

z P P i P W 0 P z = u, v

z i 2 i > 2 G[i, i + 1 , i 1, i 2,z ,u ,v ] S 1,2,3 z {u, v} S 1,1,2 + P 2

z {u, v} z i + 2 i < k 1 z P W

G W

W b V d W b = {y} y u V d G[a,b,c,d,u,y,e ] Q

W 0 V c V d W 0 V c w W 0


130/153

u V c u f G[c,b,u,w,d,e,f ] S 1,2,3 u a

G[u,w,f,e,c,b,a ] S 1,2,3 G[u,w,a,b,f,e,d ] S 1,2,3

W b, V be W b = {y} u V be y u G[b,c,a,y,u,e,f ] S 1,2,3

y u G[e,f,d,c,u,y,a ] S 1,2,3

W b = G 3 W 0 V beW 0V bV eV be{c}

V bd V d {e} {b, d}V ce V c w W 0 v V be v

V c V d v u V c f u G[v,w,e,f ,b,c ,u ] S 1,2,3

G[c,d,u,f,b,v,w ] S 1,2,3 v V c v V d

V c V d 3V b V be V bd {c} {b, e} V c V d W 0 {d} V e V ce

W b = {y} V be = V e V d u V d

v V e u a G[d,u,e,v,c,b,a ] S 1,2,3 G[d,c,e,v,u,a,y ] S 1,2,3

V e V d V b V d u V b v V d G[u,y,b,c ,v,f ,e ] S 1,2,3

3 V bV bdV d{c, e} {b, d}V eW V ce V c

G G P 6 3 3

G

(K 3, S 1,2,3, S 1,1,2 + P 2)

G G


131/153

3 (K 3, S 1,2,3, S 1,1,2 + P 2)

G P 8 C 8 G C 8 G P 8

P P

G P P

3 (K 3,S 1,2,3, S 1,1,2 + P 2) (C 7, C 8, P 8) G

G

3 |V (G)| G (K 3, S 1,2,3, S 1,1,2 +

P 2, C 7, C 8, P 8) G G

G

P 6

G P 6

Free (K 3, F ) F

m Free (K 3, mK 2)

G k k

m Free (mK 2)


132/153

mK 2 G 3 G

2 k k k

Free (mK 2) (K 3, mK 2) 2m 2

Free (K 3, mK 2) m

m Free (K 3, P 3 + mK 1)

m

Free (K 3, P 3 + mK 1)

G (K 3, P 3 + mK 1) G m

m S G R G R = V (G) S R

C = v1 v2 v p v1 p 5 S C S

vi V (C )

S

C

C v1, v2 S v1 s1 v2 s2

|S | 2 S |S | 2 m s1, v1, v2 m S \ { s1, s2} P 3 + mK 1

|S | < m + 2 G R(3, m + 2) G K 3 m + 2

C v1, v2 (P

3 + mK

1) m 1 S

K 3 C v1 v2 m 1 S |S | < 2m 1

G R(3, 2m 1)

R G R G[R] G


133/153

(P 3 + mK 1) (K 3, P 3 + mK 1)

(K 3, P 3 + mK 1)

(K 3, H )

(K 3, H )

G (K 3, H ) S G S K 1,2

x,y,z x y S N (x) S =N (y) S = {z} K 1,2

K 1,2 S {x, y}S \ { z} S

G[V \ S ] x1, . . . , x k C k 5 G[V \ S ] S S

S C S y2 S

y3 S x2 x3 x1, x2, x3, x4, y2, y3 H K 1,2 C

S x2 C

S H x1 x3 S y2 S x2 y3 S x3

x4 y2 x1, x2, y2, x3, y3, x4 H N (x2) S N (x4) S N (x4) S N (x2) S x2

x4 S x5 S

C S C S x1, x2, xk , xk 1, y1, yk

H y1 S yk S x1 xk

G


134/153

G H = L(G) K H

G v(K ) K 1, . . . , K p V (H ) {v(K 1), . . . , v (K p)}

G K 1, . . . , K p V (H )

P 6

K 1,3 + P 2

S 1,1,3

3P 2

F (K 3, F )


135/153

P 3

(net, 2K 2) (net, 4K 1)


136/153

G

ch2(G) k k G

ch2(G) 3 G G

G uv uv u

v uv uv

ch2(G) G

G ch2(G) = 2


137/153

G X G e E (G) L(e) N e GX

G[V (G) \ X ] xy E (G) x X, y V (G) \ X xy GX y

xyy xy L(xyy) = L(xy)

xyy c xy GX G

yx

w

z

G

yw

yz

w

z

G{x,y }

G GX X

a b

c

x

y

z

G

bx

cycz

x

y

z

G{a,b,c }

G GX X

G X GX G[X ] GX

G y V (G) \ X y G


138/153

GX G

x G

G e L(e) G x G

O(n + m)

G k G k

G k + 1 x y x G X = {x, y} G

GX GX k GX

G V (G) \ X GX G x y

xy G GX G y

z G yz GX z GX G yz

xy yz x y

G G GX

O(n + m)

G ch2(G) 2 G e G L(e) N

O(n + m)


139/153

e G L(e) G

G

G G C G GC G

C GC

GC G V (G) \ C G

e1, . . . , e k C

C e1 c ek e1 c

i = 2 , . . . , k ei ei 1 GC G

C

GC

G

C C GC C G

C GC G

C xy e1 ek x C, y C GC C

G c1 e1 xy e1 ek c1

i = 2 , . . . , k 1 ei ei 1 C

G


140/153


141/153

P 5

M 3

S 1,2,3

C 4


142/153

P 5


143/153

P 7

P 7

P 6

P k


144/153


145/153


146/153

d


147/153

NP

P 5

P 5

P 6


148/153


149/153


150/153

P 5

k


151/153

P 6


152/153

P P 6

P 5


153/153

P 4

Da Brow Ski Thesis

Documents

Transcript of Da Brow Ski Thesis