Da Brow Ski Thesis
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P 5
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( K r e )
(K s , K t,t ) A
M 2
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7
(K 3, F ) F
(K 3, S 1,2,3, S 1,1,2 + P 2)
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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
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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
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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
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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
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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
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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
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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
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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
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X V (G) \ X
G n f ( ) p(n) f ( ) n
p(n) 2
f (2) p(n)
P 5
L(G)2
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F F
K 3
(K 3, F ) F
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G k G k
G (G)
(G)
W [1]
> 0 n1
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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
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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
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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)
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{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
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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
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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
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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
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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
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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
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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
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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 )
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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
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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
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f (k) p(n) n f (k)
k p(n) k k
t
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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)
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(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
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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
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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
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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
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(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)
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(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
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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
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(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
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G n3 G G
G G = C 4 C 4
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G G
i(G) G
L(G)2 G
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2s 3s s
n1/ 2 > 0
k k 3
(G, k ) G k
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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 )
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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
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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
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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
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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
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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
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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 )
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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
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H k k = 3
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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
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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
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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
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d (K s , K t,t )
(K s , K t,t )
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62/153
-
8/12/2019 Da Brow Ski Thesis
63/153
G G S G S
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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
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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
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M S 1 M S 2 M S 4
S
S
M 1M 1
M 2 M 2
M 3 M 3
M 4
M 4
M 5 = M 5
= G | G
c1, c2, N nc1 n | n | nc2 n n > N n n
c > 0 |n | ncn n
k G V (G) k V 1, . . . , V k V i
G V i V j V i V j
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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
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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
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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
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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)
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(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 , )
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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
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(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}
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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
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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
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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+ , + )
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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
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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
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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
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I m C 1, . . . , C m v1, . . . , vn
J i j vi C j J i j vi C j
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
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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 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
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82/153
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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)
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n Y
n (2C 4, C 4 C 4) (2(n log
2 n ) ) (2C 4, C 4 C 4)
= M 1 M 3
-
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85/153
G G
M M
M S 1
d d 3
-
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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) =
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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
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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 |
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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)
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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
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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, . . . )
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(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
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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]
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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
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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
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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
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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})
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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]
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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
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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
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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
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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
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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
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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
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106/153
(C k , C k+1 , C k+2 , . . . ) k > 5 k = 5
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107/153
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8/12/2019 Da Brow Ski Thesis
108/153
G
G (G) G k G k k k
k k 3
4 8
3
k k
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109/153
k k
K 2,3
k k 3
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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)
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(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 )
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112/153
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8/12/2019 Da Brow Ski Thesis
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
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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 )
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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
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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
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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
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118/153
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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
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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
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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
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(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
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(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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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 )
-
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135/153
P 3
(net, 2K 2) (net, 4K 1)
-
8/12/2019 Da Brow Ski Thesis
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
-
8/12/2019 Da Brow Ski Thesis
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
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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)
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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
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140/153
-
8/12/2019 Da Brow Ski Thesis
141/153
P 5
M 3
S 1,2,3
C 4
-
8/12/2019 Da Brow Ski Thesis
142/153
P 5
-
8/12/2019 Da Brow Ski Thesis
143/153
P 7
P 7
P 6
P k
-
8/12/2019 Da Brow Ski Thesis
144/153
-
8/12/2019 Da Brow Ski Thesis
145/153
-
8/12/2019 Da Brow Ski Thesis
146/153
d
-
8/12/2019 Da Brow Ski Thesis
147/153
NP
P 5
P 5
P 6
-
8/12/2019 Da Brow Ski Thesis
148/153
-
8/12/2019 Da Brow Ski Thesis
149/153
-
8/12/2019 Da Brow Ski Thesis
150/153
P 5
k
-
8/12/2019 Da Brow Ski Thesis
151/153
P 6
-
8/12/2019 Da Brow Ski Thesis
152/153
P P 6
P 5
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153/153
P 4