Đồ thị và các thuật toán

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Mu
.
c lu
.
c
L`o
.
i n´oi d¯ˆa
`
u 7
1 D
-
a
.
i cu
.
o
.
ng vˆe
`
d¯ˆo
`
thi
.
9
1.1 D
-
i
.
nh ngh˜ıa v`a c´ac kh´ai niˆe

.
m . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 D
-
ˆo
`
thi
.
c´o hu
.
´o
.
ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 D
-
ˆo
`
thi
.
v`a ´anh xa
.
d¯a tri
.
. . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 D
-
ˆo
`
thi
.

vˆo hu
.
´o
.
ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.4 C´ac d¯i
.
nh ngh˜ıa ch´ınh . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Ma trˆa
.
n biˆe
˙’
u diˆe
˜
n d¯ˆo
`
thi
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Ma trˆa
.
n liˆen thuˆo
.

c d¯ı
˙’
nh-ca
.
nh . . . . . . . . . . . . . . . . . . . . . . 15
1.2.3 Ma trˆa
.
n kˆe
`
hay ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-d¯ı
˙’
nh . . . . . . . . . . . . . 17
1.2.4 C´ac biˆe
˙’
u diˆe
˜
n cu
˙’
a d¯ˆo
`
thi
.
. . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 T´ınh liˆen thˆong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.3.1 Dˆay chuyˆe
`
n v`a chu tr`ınh . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.2 D
-
u
.
`o
.
ng d¯i v`a ma
.
ch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.3 T´ınh liˆen thˆong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1

1.3.4 Cˆa
`
u, k−liˆen thˆong . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.5 D
-
ˆo
`
thi
.
liˆen thˆong ma
.
nh . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4 Pha
.
m vi v`a liˆen thˆong ma

.
nh . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.1 Ma trˆa
.
n pha
.
m vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.2 T`ım c´ac th`anh phˆa
`
n liˆen thˆong ma
.
nh . . . . . . . . . . . . . . . . . . 36
1.4.3 Co
.
so
.
˙’
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5 D
-
ˇa
˙’
ng cˆa
´
u cu
˙’
a c´ac d¯ˆo
`
thi
.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5.1 1−d¯ˇa
˙’
ng cˆa
´
u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.5.2 2−d¯ˇa
˙’
ng cˆa
´
u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.6 C´ac d¯ˆo
`
thi
.
d¯ˇa
.
c biˆe
.
t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.6.1 D
-
ˆo
`
thi
.
khˆong c´o ma
.
ch . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.6.2 D

-
ˆo
`
thi
.
phˇa
˙’
ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2 C´ac sˆo
´
co
.
ba
˙’
n cu
˙’
a d¯ˆo
`
thi
.
49
2.1 Chu sˆo
´
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 Sˇa
´
c sˆo
´
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2.1 C´ach t`ım sˇa

´
c sˆo
´
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Sˆo
´
ˆo
˙’
n d¯i
.
nh trong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 Sˆo
´
ˆo
˙’
n d¯i
.
nh ngo`ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.5 Phu
˙’
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6 Nhˆan cu
˙’
a d¯ˆo
`
thi
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6.1 C´ac d¯i
.

nh l´y vˆe
`
tˆo
`
n ta
.
i v`a duy nhˆa
´
t . . . . . . . . . . . . . . . . . . . 69
2.6.2 Tr`o cho
.
i Nim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2

3 C´ac b`ai to´an vˆe
`
d¯u
.
`o
.
ng d¯i 75
3.1 D
-
u
.
`o
.
ng d¯i gi˜u
.
a hai d¯ı

˙’
nh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1.1 D
-
u
.
`o
.
ng d¯i gi˜u
.
a hai d¯ı
˙’
nh . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1.2 D
-
ˆo
`
thi
.
liˆen thˆong ma
.
nh . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 D
-
u
.
`o
.
ng d¯i ngˇa
´

n nhˆa
´
t gi˜u
.
a hai d¯ı
˙’
nh . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.1 Tru
.
`o
.
ng ho
.
.
p ma trˆa
.
n tro
.
ng lu
.
o
.
.
ng khˆong ˆam . . . . . . . . . . . . . . 78
3.2.2 Tru
.
`o
.
ng ho
.

.
p ma trˆa
.
n tro
.
ng lu
.
o
.
.
ng tu`y ´y . . . . . . . . . . . . . . . . . 82
3.3 D
-
u
.
`o
.
ng d¯i ngˇa
´
n nhˆa
´
t gi˜u
.
a tˆa
´
t ca
˙’
c´ac cˇa
.
p d¯ı

˙’
nh . . . . . . . . . . . . . . . . . 87
3.3.1 Thuˆa
.
t to´an Hedetniemi (tru
.
`o
.
ng ho
.
.
p ma trˆa
.
n tro
.
ng lu
.
o
.
.
ng khˆong ˆam) 88
3.3.2 Thuˆa
.
t to´an Floyd (tru
.
`o
.
ng ho
.
.

p ma trˆa
.
n tro
.
ng lu
.
o
.
.
ng tu`y ´y) . . . . . . 93
3.4 Ph´at hiˆe
.
n ma
.
ch c´o d¯ˆo
.
d`ai ˆam . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.4.1 Ma
.
ch tˆo
´
i u
.
u trong d¯ˆo
`
thi
.
c´o hai tro
.
ng lu

.
o
.
.
ng . . . . . . . . . . . . . . 96
4 C
ˆ
AY 99
4.1 Mo
.
˙’
d¯ˆa
`
u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Cˆay Huﬀman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.1 C´ac bˆo
.
m˜a “tˆo
´
t” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.2 M˜a Huﬀman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Cˆay bao tr`um . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.1 Thuˆa
.
t to´an t`ım kiˆe
´
m theo chiˆe
`
u rˆo
.

ng x´ac d¯i
.
nh cˆay bao tr`um . . . . . 107
4.3.2 Thuˆa
.
t to´an t`ım kiˆe
´
m theo chiˆe
`
u sˆau x´ac d¯i
.
nh cˆay bao tr`um . . . . . 107
4.3.3 T`ım cˆay bao tr`um du
.
.
a trˆen hai ma
˙’
ng tuyˆe
´
n t´ınh . . . . . . . . . . . 108
4.3.4 Thuˆa
.
t to´an t`ım tˆa
´
t ca
˙’
c´ac cˆay bao tr`um . . . . . . . . . . . . . . . . 112
4.3.5 Hˆe
.
co

.
so
.
˙’
cu
˙’
a c´ac chu tr`ınh d¯ˆo
.
c lˆa
.
p . . . . . . . . . . . . . . . . . . . 112
3

4.4 Cˆay bao tr`um tˆo
´
i thiˆe
˙’
u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4.1 Thuˆa
.
t to´an Kruskal . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.2 Thuˆa
.
t to´an Prim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4.3 Thuˆa
.
t to´an Dijkstra-Kevin-Whitney . . . . . . . . . . . . . . . . . . 121
4.5 B`ai to´an Steiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5 B`ai to´an Euler v`a b`ai to´an Hamilton 127
5.1 B`ai to´an Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.1.1 Thuˆa
.
t to´an t`ım dˆay chuyˆe
`
n Euler . . . . . . . . . . . . . . . . . . . . 129
5.2 B`ai to´an ngu
.
`o
.
i d¯u
.
a thu
.
Trung Hoa . . . . . . . . . . . . . . . . . . . . . . . 131
5.3 B`ai to´an Hamilton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.3.1 C´ac d¯iˆe
`
u kiˆe
.
n cˆa
`
n d¯ˆe
˙’
tˆo
`
n ta
.
i chu tr`ınh Hamilton . . . . . . . . . . . . 138
5.3.2 C´ac d¯iˆe
`

u kiˆe
.
n d¯u
˙’
vˆe
`
su
.
.
tˆo
`
n ta
.
i chu tr`ınh Hamilton . . . . . . . . . . 139
5.3.3 C´ac d¯iˆe
`
u kiˆe
.
n d¯u
˙’
vˆe
`
su
.
.
tˆo
`
n ta
.
i ma

.
ch Hamilton . . . . . . . . . . . . . 142
6 D
-
ˆo
`
thi
.
phˇa
˙’
ng 149
6.1 D
-
i
.
nh ngh˜ıa v`a c´ac v´ı du
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 C´ac biˆe
˙’
u diˆe
˜
n kh´ac nhau cu
˙’
a mˆo
.
t d¯ˆo
`
thi
.

phˇa
˙’
ng . . . . . . . . . . . . . . . . 151
6.3 C´ac t´ınh chˆa
´
t cu
˙’
a d¯ˆo
`
thi
.
phˇa
˙’
ng . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.4 Ph´at hiˆe
.
n t´ınh phˇa
˙’
ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.4.1 Kiˆe
˙’
m tra t´ınh phˇa
˙’
ng . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.5 D
-
ˆo
´
i ngˆa
˜

u h`ınh ho
.
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.6 D
-
ˆo
´
i ngˆa
˜
u tˆo
˙’
ho
.
.
p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7 Ma
.
ng vˆa
.
n ta
˙’
i 173
4

7.1 Mo
.
˙’
d¯ˆa
`
u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.2 B`ai to´an luˆo
`
ng l´o
.
n nhˆa
´
t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.2.1 Thuˆa
.
t to´an g´an nh˜an d¯ˆe
˙’
t`ım luˆo
`
ng l´o
.
n nhˆa
´
t . . . . . . . . . . . . . . 180
7.2.2 D
-
ˆo
`
thi
.
d¯iˆe
`
u chı
˙’
nh luˆo
`

ng . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.2.3 Phˆan t´ıch luˆo
`
ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.3 C´ac ca
˙’
i biˆen d¯o
.
n gia
˙’
n cu
˙’
a b`ai to´an luˆo
`
ng l´o
.
n nhˆa
´
t . . . . . . . . . . . . . . . 183
7.3.1 C´ac d¯ˆo
`
thi
.
c´o nhiˆe
`
u nguˆo
`
n v`a nhiˆe
`
u d¯´ıch . . . . . . . . . . . . . . . . 183

7.3.2 C´ac d¯ˆo
`
thi
.
v´o
.
i r`ang buˆo
.
c ta
.
i c´ac cung v`a d¯ı
˙’
nh . . . . . . . . . . . . . 184
7.3.3 C´ac d¯ˆo
`
thi
.
c´o cˆa
.
n trˆen v`a cˆa
.
n du
.
´o
.
i vˆe
`
luˆo
`
ng . . . . . . . . . . . . . . 185

7.4 Luˆo
`
ng v´o
.
i chi ph´ı nho
˙’
nhˆa
´
t . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.4.1 Thuˆa
.
t to´an Klein, Busacker, Gowen . . . . . . . . . . . . . . . . . . . 186
7.5 Cˇa
.
p gh´ep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.5.1 C´ac b`ai to´an vˆe
`
cˇa
.
p gh´ep . . . . . . . . . . . . . . . . . . . . . . . . 189
7.5.2 Cˇa
.
p gh´ep l´o
.
n nhˆa
´
t trong d¯ˆo
`
thi
.

hai phˆa
`
n . . . . . . . . . . . . . . . . 192
7.5.3 Cˇa
.
p gh´ep ho`an ha
˙’
o trong d¯ˆo
`
thi
.
hai phˆa
`
n . . . . . . . . . . . . . . . 193
A Thu
.
viˆe
.
n Graph.h 197
T`ai liˆe
.
u tham kha
˙’
o 209
5

6

L`o
.

i n´oi d¯ˆa
`
u
Trong thu
.
.
c tˆe
´
d¯ˆe
˙’
miˆeu ta
˙’
mˆo
.
t sˆo
´
t`ınh huˆo
´
ng ngu
.
`o
.
i ta thu
.
`o
.
ng biˆe
˙’
u thi
.

bˇa
`
ng mˆo
.
t h`ınh a
˙’
nh
gˆo
`
m c´ac d¯iˆe
˙’
m (c´ac d¯ı
˙’
nh)-biˆe
˙’
u diˆe
˜
n c´ac thu
.
.
c thˆe
˙’
-v`a v˜e c´ac d¯oa
.
n thˇa
˙’
ng nˆo
´
i cˇa
.

p c´ac d¯ı
˙’
nh biˆe
˙’
u
diˆe
˜
n mˆo
´
i quan hˆe
.
gi˜u
.
a ch´ung. Nh˜u
.
ng h`ınh nhu
.
thˆe
´
thu
.
`o
.
ng go
.
i l`a c´ac d¯ˆo
`
thi
.
. Mu

.
c d¯´ıch cu
˙’
a
gi´ao tr`ınh n`ay cung cˆa
´
p nh˜u
.
ng kiˆe
´
n th´u
.
c co
.
ba
˙’
n d¯ˆe
˙’
nghiˆen c´u
.
u c´ac d¯ˆo
`
thi
.
. C´ac d¯ˆo
`
thi
.
xuˆa
´

t
hiˆe
.
n trong nhiˆe
`
u l˜ınh vu
.
.
c v´o
.
i c´ac tˆen go
.
i kh´ac nhau: “cˆa
´
u tr´uc” trong cˆong tr`ınh xˆay du
.
.
ng,
“ma
.
ch” trong d¯iˆe
.
n tu
.
˙’
, “lu
.
o
.
.

c d¯ˆo
`
quan hˆe
.
”, “cˆa
´
u tr´uc truyˆe
`
n thˆong”, “cˆa
´
u tr´uc tˆo
˙’
ch´u
.
c”
trong x˜a hˆo
.
i v`a kinh tˆe
´
, “cˆa
´
u tr´uc phˆan tu
.
˙’
” trong ho´a ho
.
c, vˆan vˆan.
Do nh˜u
.
ng ´u

.
ng du
.
ng rˆo
.
ng r˜ai cu
˙’
a n´o trong nhiˆe
`
u l˜ınh vu
.
.
c, c´o rˆa
´
t nhiˆe
`
u nghiˆen c´u
.
u
xung quanh l´y thuyˆe
´
t d¯ˆo
`
thi
.
trong nh˜u
.
ng nˇam gˆa
`
n d¯ˆay; mˆo

.
t nhˆan tˆo
´
chu
˙’
yˆe
´
u g´op phˆa
`
n th´uc
d¯ˆa
˙’
y su
.
.
ph´at triˆe
˙’
n d¯´o l`a xuˆa
´
t hiˆe
.
n c´ac m´ay t´ınh l´o
.
n c´o thˆe
˙’
thu
.
.
c hiˆe
.

n nhiˆe
`
u ph´ep to´an v´o
.
i
tˆo
´
c d¯ˆo
.
rˆa
´
t nhanh. Viˆe
.
c biˆe
˙’
u diˆe
˜
n tru
.
.
c tiˆe
´
p v`a chi tiˆe
´
t c´ac hˆe
.
thˆo
´
ng thu
.

.
c tˆe
´
, chˇa
˙’
ng ha
.
n c´ac
ma
.
ng truyˆe
`
n thˆong, d¯˜a d¯u
.
a d¯ˆe
´
n nh˜u
.
ng d¯ˆo
`
thi
.
c´o k´ıch thu
.
´o
.
c l´o
.
n v`a viˆe
.

c phˆan t´ıch th`anh
cˆong hˆe
.
thˆo
´
ng phu
.
thuˆo
.
c rˆa
´
t nhiˆe
`
u v`ao c´ac thuˆa
.
t to´an “tˆo
´
t” c˜ung nhu
.
kha
˙’
nˇang cu
˙’
a m´ay
t´ınh. Theo d¯´o, gi´ao tr`ınh n`ay s˜e tˆa
.
p trung v`ao viˆe
.
c ph´at triˆe
˙’

n v`a tr`ınh b`ay c´ac thuˆa
.
t to´an
d¯ˆe
˙’
phˆan t´ıch c´ac d¯ˆo
`
thi
.
.
C´ac phu
.
o
.
ng ph´ap phˆan t´ıch v`a thiˆe
´
t kˆe
´
c´ac thuˆa
.
t to´an trong gi´ao tr`ınh cho ph´ep sinh
viˆen c´o thˆe
˙’
viˆe
´
t dˆe
˜
d`ang c´ac chu
.
o

.
ng tr`ınh minh ho
.
a. Gi´ao tr`ınh d¯u
.
o
.
.
c biˆen soa
.
n cho c´ac d¯ˆo
´
i
tu
.
o
.
.
ng l`a sinh viˆen To´an-Tin v`a Tin ho
.
c.
Gi´ao tr`ınh su
.
˙’
du
.
ng ngˆon ng˜u
.
C d¯ˆe
˙’

minh ho
.
a, tuy nhiˆen c´o thˆe
˙’
dˆe
˜
d`ang chuyˆe
˙’
n d¯ˆo
˙’
i
sang c´ac ngˆon ng˜u
.
kh´ac; v`a do d¯´o, sinh viˆen cˆa
`
n c´o mˆo
.
t sˆo
´
kiˆe
´
n th´u
.
c vˆe
`
ngˆon ng˜u
.
C. Ngo`ai
ra, hˆa
`

u hˆe
´
t c´ac chu
.
o
.
ng tr`ınh thao t´ac trˆen cˆa
´
u tr´uc d˜u
.
liˆe
.
u nhu
.
danh s´ach liˆen kˆe
´
t, nˆen d¯`oi
ho
˙’
i sinh viˆen pha
˙’
i c´o nh˜u
.
ng k˜y nˇang lˆa
.
p tr`ınh tˆo
´
t.
Gi´ao tr`ınh bao gˆo
`

m ba
˙’
y chu
.
o
.
ng v`a mˆo
.
t phˆa
`
n phu
.
lu
.
c v´o
.
i nh˜u
.
ng nˆo
.
i dung ch´ınh nhu
.
sau:
• Chu
.
o
.
ng th´u
.
nhˆa

´
t tr`ınh b`ay nh˜u
.
ng kh´ai niˆe
.
m cˇan ba
˙’
n vˆe
`
d¯ˆo
`
thi
.
.
• Chu
.
o
.
ng 2 tr`ınh b`ay nh˜u
.
ng sˆo
´
co
.
ba
˙’
n cu
˙’
a d¯ˆo
`

thi
.
.
´
Y ngh˜ıa thu
.
.
c tiˆe
˜
n cu
˙’
a c´ac sˆo
´
n`ay.
7

• Chu
.
o
.
ng 3 t`ım hiˆe
˙’
u b`ai to´an t`ım d¯u
.
`o
.
ng d¯i ngˇa
´
n nhˆa
´

t.
• Chu
.
o
.
ng 4 d¯ˆe
`
cˆa
.
p d¯ˆe
´
n kh´ai niˆe
.
m vˆe
`
cˆay.
´
U
.
ng du
.
ng cu
˙’
a cˆay Huﬀman trong n´en d˜u
.
liˆe
.
u. Ngo`ai ra xˆay du
.
.

ng c´ac thuˆa
.
t to´an t`ım cˆay bao tr`um nho
˙’
nhˆa
´
t.
• B`ai to´an Euler v`a b`ai to´an Hamilton v`a nh˜u
.
ng mo
.
˙’
rˆo
.
ng cu
˙’
a ch´ung s˜e d¯u
.
o
.
.
c n´oi d¯ˆe
´
n
trong Chu
.
o
.
ng 5.
• Chu

.
o
.
ng 6 nghiˆen c´u
.
u c´ac t´ınh chˆa
´
t phˇa
˙’
ng cu
˙’
a d¯ˆo
`
thi
.
; v`a cuˆo
´
i c`ung
• Chu
.
o
.
ng 7 t`ım hiˆe
˙’
u c´ac b`ai to´an trˆen ma
.
ng vˆa
.
n ta
˙’

i.
Ngo`ai ra, phˆa
`
n phu
.
lu
.
c tr`ınh b`ay c´ac cˆa
´
u tr´uc d˜u
.
liˆe
.
u v`a nh˜u
.
ng thu
˙’
tu
.
c cˆa
`
n thiˆe
´
t d¯ˆe
˙’
d¯o
.
n gia
˙’
n ho´a c´ac d¯oa

.
n chu
.
o
.
ng tr`ınh minh ho
.
a c´ac thuˆa
.
t to´an d¯u
.
o
.
.
c tr`ınh b`ay.
Gi´ao tr`ınh d¯u
.
o
.
.
c biˆen soa
.
n lˆa
`
n d¯ˆa
`
u tiˆen nˆen khˆong tr´anh kho
˙’
i kh´a nhiˆe
`

u thiˆe
´
u s´ot. T´ac
gia
˙’
mong c´o nh˜u
.
ng d¯´ong g´op t`u
.
ba
.
n d¯o
.
c.
Tˆoi xin ca
˙’
m o
.
n nh˜u
.
ng gi´up d¯˜o
.
d¯˜a nhˆa
.
n d¯u
.
o
.
.
c t`u

.
nhiˆe
`
u ngu
.
`o
.
i m`a khˆong thˆe
˙’
liˆe
.
t kˆe
hˆe
´
t, d¯ˇa
.
c biˆe
.
t l`a c´ac ba
.
n sinh viˆen, trong qu´a tr`ınh biˆen soa
.
n gi´ao tr`ınh n`ay.
D
-
`a La
.
t, ng`ay 5 th´ang 3 nˇam 2002
PHA
.

M Tiˆe
´
n So
.
n
8

Chu
.
o
.
ng 1
D
-
a
.
i cu
.
o
.
ng vˆe
`
d¯ˆo
`
thi
.
1.1 D
-
i
.

nh ngh˜ıa v`a c´ac kh´ai niˆe
.
m
1.1.1 D
-
ˆo
`
thi
.
c´o hu
.
´o
.
ng
D
-
ˆo
`
thi
.
c´o hu
.
´o
.
ng G = (V, E) gˆo
`
m mˆo
.
t tˆa
.

p V c´ac phˆa
`
n tu
.
˙’
go
.
i l`a d¯ı
˙’
nh (hay n´ut) v`a mˆo
.
t tˆa
.
p
E c´ac cung sao cho mˆo
˜
i cung e ∈ E tu
.
o
.
ng ´u
.
ng v´o
.
i mˆo
.
t cˇa
.
p c´ac d¯ı
˙’

nh d¯u
.
o
.
.
c sˇa
´
p th´u
.
tu
.
.
. Nˆe
´
u
c´o d¯´ung mˆo
.
t cung e tu
.
o
.
ng ´u
.
ng c´ac d¯ı
˙’
nh d¯u
.
o
.
.

c sˇa
´
p th´u
.
tu
.
.
(a, b), ta s˜e viˆe
´
t e := (a, b).
Ch´ung ta s˜e gia
˙’
su
.
˙’
c´ac d¯ı
˙’
nh d¯u
.
o
.
.
c d¯´anh sˆo
´
l`a v
1
, v
2
, . . . , v
n

hay gia
˙’
n tiˆe
.
n, 1, 2, . . . , n,
trong d¯´o n = #V l`a sˆo
´
c´ac d¯ı
˙’
nh cu
˙’
a d¯ˆo
`
thi
.
.
Nˆe
´
u e l`a mˆo
.
t cung tu
.
o
.
ng ´u
.
ng cˇa
.
p c´ac d¯ı
˙’

nh d¯u
.
o
.
.
c sˇa
´
p th´u
.
tu
.
.
v
i
v`a v
j
th`ı d¯ı
˙’
nh v
i
go
.
i l`a
gˆo
´
c v`a d¯ı
˙’
nh v
j
go

.
i l`a ngo
.
n; cung e go
.
i l`a liˆen thuˆo
.
c hai d¯ı
˙’
nh v
i
v`a v
j
. Ch´ung ta s˜e thu
.
`o
.
ng
k´y hiˆe
.
u m = #E−sˆo
´
ca
.
nh cu
˙’
a d¯ˆo
`
thi
.

G. C´ac ca
.
nh thu
.
`o
.
ng d¯u
.
o
.
.
c d¯´anh sˆo
´
l`a e
1
, e
2
, . . . , e
m
.
Mˆo
.
t c´ach h`ınh ho
.
c, c´ac d¯ı
˙’
nh d¯u
.
o
.

.
c biˆe
˙’
u diˆe
˜
n bo
.
˙’
i c´ac d¯iˆe
˙’
m, v`a e = (v
i
, v
j
) d¯u
.
o
.
.
c biˆe
˙’
u
diˆe
˜
n bo
.
˙’
i mˆo
.
t cung nˆo

´
i c´ac d¯iˆe
˙’
m v
i
v`a v
j
.
Mˆo
.
t cung c´o gˆo
´
c tr`ung v´o
.
i ngo
.
n go
.
i l`a khuyˆen.
Nˆe
´
u c´o nhiˆe
`
u ho
.
n mˆo
.
t cung v´o
.
i gˆo

´
c ta
.
i v
i
v`a ngo
.
n ta
.
i v
j
th`ı G go
.
i l`a d¯a d¯ˆo
`
thi
.
v`a c´ac
cung tu
.
o
.
ng ´u
.
ng go
.
i l`a song song. D
-
o
.

n d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng l`a d¯ˆo
`
thi
.
khˆong khuyˆen trong d¯´o hai
d¯ı
˙’
nh bˆa
´
t k`y v
i
v`a v
j
c´o nhiˆe
`
u nhˆa
´
t mˆo
.
t cung (v
i
, v

j
). Chˇa
˙’
ng ha
.
n, d¯ˆo
`
thi
.
trong H`ınh 1.1 c´o
cung e
8
l`a khuyˆen; c´ac cung e
4
v`a e
9
l`a song song do c`ung tu
.
o
.
ng ´u
.
ng cˇa
.
p d¯ı
˙’
nh v
3
v`a v
4

.
9

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.....................................
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v
1
v
2

v
3
v
4
v
5
e
1
e
2
e
3
e
4
e
5
e
6
e
7
e
8
e
9
•
• •
•
•
H`ınh 1.1: V´ı du
.

cu
˙’
a 2−d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng.
1.1.2 D
-
ˆo
`
thi
.
v`a ´anh xa
.
d¯a tri
.
V´o
.
i mˆo
˜
i x ∈ V, k´y hiˆe
.
u Γ(x) := {y ∈ V | (x, y) ∈ E}. Khi d¯´o ta c´o mˆo
.
t ´anh xa

.
d¯a tri
.
Γ: V → 2
V
, x → Γ(x). K´y hiˆe
.
u Γ
−1
l`a ´anh xa
.
(d¯a tri
.
) ngu
.
o
.
.
c cu
˙’
a Γ.
Nˆe
´
u G l`a d¯o
.
n d¯ˆo
`
thi
.
, th`ı d¯ˆo

`
thi
.
n`ay ho`an to`an d¯u
.
o
.
.
c x´ac d¯i
.
nh bo
.
˙’
i tˆa
.
p V v`a ´anh xa
.
d¯a
tri
.
Γ t`u
.
V v`ao 2
V
. V`ı vˆa
.
y, d¯ˆo
`
thi
.

n`ay c`on c´o thˆe
˙’
k´y hiˆe
.
u l`a G = (V, Γ).
Nˆe
´
u xo´a cung e
9
trong H`ınh 1.1 ta nhˆa
.
n d¯u
.
o
.
.
c d¯o
.
n d¯ˆo
`
thi
.
v`a do d¯´o c´o thˆe
˙’
biˆe
˙’
u diˆe
˜
n
bo

.
˙’
i ´anh xa
.
d¯a tri
.
Γ. Trong tru
.
`o
.
ng ho
.
.
p n`ay ta c´o
Γ(v
1
) = {v
2
}, Γ(v
2
) = {v
1
, v
3
}, Γ(v
3
) = {v
4
, v
5

}, Γ(v
4
) = {v
5
}, Γ(v
5
) = {v
1
, v
5
}.
1.1.3 D
-
ˆo
`
thi
.
vˆo hu
.
´o
.
ng
Khi nghiˆen c´u
.
u mˆo
.
t sˆo
´
t´ınh chˆa
´

t cu
˙’
a c´ac d¯ˆo
`
thi
.
, ta thˆa
´
y rˇa
`
ng ch´ung khˆong phu
.
thuˆo
.
c v`ao
hu
.
´o
.
ng cu
˙’
a c´ac cung, t´u
.
c l`a khˆong cˆa
`
n phˆan biˆe
.
t su
.
.

kh´ac nhau gi˜u
.
a c´ac d¯iˆe
˙’
m bˇa
´
t d¯ˆa
`
u v`a
kˆe
´
t th´uc. D
-
iˆe
`
u n`ay d¯o
.
n gia
˙’
n l`a mˆo
˜
i khi c´o ´ıt nhˆa
´
t mˆo
.
t cung gi˜u
.
a hai d¯ı
˙’
nh ta khˆong quan

tˆam d¯ˆe
´
n th´u
.
tu
.
.
cu
˙’
a ch´ung.
V´o
.
i mˆo
˜
i cung, t´u
.
c l`a mˆo
˜
i cˇa
.
p c´o th´u
.
tu
.
.
(v
i
, v
j
) ta cho tu

.
o
.
ng ´u
.
ng cˇa
.
p khˆong c´o th´u
.
tu
.
.
(v
i
, v
j
) go
.
i l`a c´ac ca
.
nh. Tu
.
o
.
ng d¯u
.
o
.
ng, ta n´oi rˇa
`

ng ca
.
nh l`a mˆo
.
t cung m`a hu
.
´o
.
ng d¯˜a bi
.
bo
˙’
quˆen. Vˆe
`
h`ınh ho
.
c, ca
.
nh (v
i
, v
j
) d¯u
.
o
.
.
c biˆe
˙’
u diˆe

˜
n bo
.
˙’
i c´ac d¯oa
.
n thˇa
˙’
ng (hoˇa
.
c cong) v`a khˆong
c´o m˜ui tˆen liˆen thuˆo
.
c hai d¯iˆe
˙’
m tu
.
o
.
ng ´u
.
ng hai d¯ı
˙’
nh v
i
v`a v
j
.
10

Nghiˆen c´u
.
u c´ac t´ınh chˆa
´
t vˆo hu
.
´o
.
ng cu
˙’
a d¯ˆo
`
thi
.
G = (V, E) d¯u
.
a vˆe
`
kha
˙’
o s´at tˆa
.
p E l`a
tˆa
.
p c´ac ca
.
nh, t´u
.
c l`a, mˆo

.
t tˆa
.
p h˜u
.
u ha
.
n c´ac phˆa
`
n tu
.
˙’
m`a mˆo
˜
i phˆa
`
n tu
.
˙’
l`a mˆo
.
t cˇa
.
p hai d¯ı
˙’
nh
phˆan biˆe
.
t hay d¯ˆo
`

ng nhˆa
´
t cu
˙’
a V.
D
-
a d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng l`a d¯ˆo
`
thi
.
m`a c´o thˆe
˙’
c´o nhiˆe
`
u ho
.
n mˆo
.
t ca
.
nh liˆen thuˆo

.
c hai d¯ı
˙’
nh.
D
-
ˆo
`
thi
.
go
.
i l`a d¯o
.
n nˆe
´
u n´o khˆong c´o khuyˆen v`a hai d¯ı
˙’
nh bˆa
´
t k`y c´o nhiˆe
`
u nhˆa
´
t mˆo
.
t ca
.
nh
liˆen thuˆo

.
c ch´ung.
................................................................................................................................................................................................................................................................................................
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v
1
v
2
v
3
v
4
v
5
e
1
e
2
e
3

e
4
e
5
e
6
e
7
e
8
e
9
•
• •
•
•
H`ınh 1.2: D
-
ˆo
`
thi
.
vˆo hu
.
´o
.
ng tu
.
o
.

ng ´u
.
ng d¯ˆo
`
thi
.
trong H`ınh 1.1.
1.1.4 C´ac d¯i
.
nh ngh˜ıa ch´ınh
Hai cung, hoˇa
.
c hai ca
.
nh go
.
i l`a kˆe
`
nhau nˆe
´
u ch´ung c´o ´ıt nhˆa
´
t mˆo
.
t d¯ı
˙’
nh chung. Chˇa
˙’
ng ha
.

n,
hai ca
.
nh e
1
v`a e
3
trong H`ınh 1.2 l`a kˆe
`
nhau. Hai d¯ı
˙’
nh v
i
v`a v
j
go
.
i l`a kˆe
`
nhau nˆe
´
u tˆo
`
n ta
.
i
ca
.
nh hoˇa
.

c cung e
k
liˆen thuˆo
.
c ch´ung. V´ı du
.
trong H`ınh 1.2 hai d¯ı
˙’
nh v
2
v`a v
3
l`a kˆe
`
nhau (liˆen
thuˆo
.
c bo
.
˙’
i ca
.
nh e
3
), nhu
.
ng d¯ı
˙’
nh v
2

v`a v
5
khˆong kˆe
`
nhau.
Bˆa
.
c v`a nu
.
˙’
a bˆa
.
c
Bˆa
.
c ngo`ai cu
˙’
a d¯ı
˙’
nh v ∈ V, k´y hiˆe
.
u d
+
G
(v) (hay d
+
(v) nˆe
´
u khˆong so
.

.
nhˆa
`
m lˆa
˜
n) l`a sˆo
´
c´ac cung
c´o d¯ı
˙’
nh v l`a gˆo
´
c. Bˆa
.
c trong cu
˙’
a d¯ı
˙’
nh v ∈ V, k´y hiˆe
.
u d
−
G
(v) (hay d
−
(v) nˆe
´
u khˆong so
.
.

nhˆa
`
m
lˆa
˜
n) l`a sˆo
´
c´ac cung c´o d¯ı
˙’
nh v l`a ngo
.
n.
Chˇa
˙’
ng ha
.
n, d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng trong H`ınh 1.1 c´o d
+
(v
2
) = 2, d
−

(v
2
) = 1.
11

Hiˆe
˙’
n nhiˆen rˇa
`
ng, tˆo
˙’
ng c´ac bˆa
.
c ngo`ai cu
˙’
a c´ac d¯ı
˙’
nh bˇa
`
ng tˆo
˙’
ng c´ac bˆa
.
c trong cu
˙’
a c´ac
d¯ı
˙’
nh v`a bˇa
`

ng tˆo
˙’
ng sˆo
´
cung cu
˙’
a d¯ˆo
`
thi
.
G, t´u
.
c l`a
n

i=1
d
+
(v
i
) =
n

i=1
d
−
(v
i
) = m.
Nˆe

´
u G l`a d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng, bˆa
.
c cu
˙’
a d¯ı
˙’
nh v ∈ V, k´y hiˆe
.
u d
G
(v) (hay d(v) nˆe
´
u khˆong so
.
.
nhˆa
`
m lˆa
˜
n) l`a sˆo
´

c´ac ca
.
nh liˆen thuˆo
.
c d¯ı
˙’
nh v v´o
.
i khuyˆen d¯u
.
o
.
.
c d¯ˆe
´
m hai lˆa
`
n. V´ı du
.
d¯ˆo
`
thi
.
vˆo
hu
.
´o
.
ng trong H`ınh 1.2 c´o d(v
2

) = 3, d(v
5
) = 5.
C´ac cung (ca
.
nh) liˆen thuˆo
.
c tˆa
.
p A ⊂ V. C´ac d¯ˆo
´
i chu tr`ınh
Gia
˙’
su
.
˙’
A ⊂ V. K´y hiˆe
.
u ω
+
(A) l`a tˆa
.
p tˆa
´
t ca
˙’
c´ac cung c´o d¯ı
˙’
nh gˆo

´
c thuˆo
.
c A v`a d¯ı
˙’
nh ngo
.
n
thuˆo
.
c A
c
:= V \ A, v`a ω
−
(A) l`a tˆa
.
p tˆa
´
t ca
˙’
c´ac cung c´o d¯ı
˙’
nh ngo
.
n thuˆo
.
c A v`a d¯ı
˙’
nh gˆo
´

c thuˆo
.
c
A
c
. D
-
ˇa
.
t
ω(A) = ω
+
(A) ∪ ω
−
(A).
Tˆa
.
p c´ac cung hoˇa
.
c ca
.
nh c´o da
.
ng ω(A) go
.
i l`a d¯ˆo
´
i chu tr`ınh cu
˙’
a d¯ˆo

`
thi
.
.
D
-
ˆo
`
thi
.
c´o tro
.
ng sˆo
´
D
-
ˆo
`
thi
.
c´o tro
.
ng sˆo
´
nˆe
´
u trˆen mˆo
˜
i cung (hoˇa
.

c ca
.
nh) e ∈ E c´o tu
.
o
.
ng ´u
.
ng mˆo
.
t sˆo
´
thu
.
.
c w(e) go
.
i
l`a tro
.
ng lu
.
o
.
.
ng cu
˙’
a cung e.
D
-

ˆo
`
thi
.
d¯ˆo
´
i x´u
.
ng
D
-
ˆo
`
thi
.
c´o hu
.
´o
.
ng go
.
i l`a d¯ˆo
´
i x´u
.
ng nˆe
´
u c´o bao nhiˆeu cung da
.
ng (v

i
, v
j
) th`ı c˜ung c´o bˆa
´
y nhiˆeu
cung da
.
ng (v
j
, v
i
).
D
-
ˆo
`
thi
.
pha
˙’
n d¯ˆo
´
i x´u
.
ng
D
-
ˆo
`

thi
.
c´o hu
.
´o
.
ng go
.
i l`a pha
˙’
n d¯ˆo
´
i x´u
.
ng nˆe
´
u c´o cung da
.
ng (v
i
, v
j
) th`ı khˆong c´o cung da
.
ng
(v
j
, v
i
).

12

D
-
ˆo
`
thi
.
d¯ˆa
`
y d¯u
˙’
D
-
ˆo
`
thi
.
vˆo hu
.
´o
.
ng go
.
i l`a d¯ˆa
`
y d¯u
˙’
nˆe
´

u hai d¯ı
˙’
nh bˆa
´
t k`y v
i
v`a v
j
tˆo
`
n ta
.
i mˆo
.
t ca
.
nh da
.
ng (v
i
, v
j
).
D
-
o
.
n d¯ˆo
`
thi

.
vˆo hu
.
´o
.
ng d¯ˆa
`
y d¯u
˙’
n d¯ı
˙’
nh d¯u
.
o
.
.
c k´y hiˆe
.
u l`a K
n
.
D
-
ˆo
`
thi
.
con
Gia
˙’

su
.
˙’
A ⊂ V. D
-
ˆo
`
thi
.
con d¯u
.
o
.
.
c sinh bo
.
˙’
i tˆa
.
p A l`a d¯ˆo
`
thi
.
G
A
:= (A, E
A
) trong d¯´o c´ac d¯ı
˙’
nh l`a

c´ac phˆa
`
n tu
.
˙’
cu
˙’
a tˆa
.
p A v`a c´ac cung trong E
A
l`a c´ac cung cu
˙’
a G m`a hai d¯ı
˙’
nh n´o liˆen thuˆo
.
c
thuˆo
.
c tˆa
.
p A.
Nˆe
´
u G l`a d¯ˆo
`
thi
.
biˆe

˙’
u diˆe
˜
n ba
˙’
n d¯ˆo
`
giao thˆong cu
˙’
a nu
.
´o
.
c Viˆe
.
t Nam th`ı d¯ˆo
`
thi
.
biˆe
˙’
u diˆe
˜
n
ba
˙’
n d¯ˆo
`
giao thˆong cu
˙’

a th`anh phˆo
´
D
-
`a La
.
t l`a mˆo
.
t d¯ˆo
`
thi
.
con.
D
-
ˆo
`
thi
.
bˆo
.
phˆa
.
n
X´et d¯ˆo
`
thi
.
G = (V, E) v`a U ⊂ E. D
-

ˆo
`
thi
.
bˆo
.
phˆa
.
n sinh bo
.
˙’
i tˆa
.
p U l`a d¯ˆo
`
thi
.
v´o
.
i tˆa
.
p d¯ı
˙’
nh V
v`a c´ac cung thuˆo
.
c U (c´ac cung cu
˙’
a E \ U bi
.

xo´a kho
˙’
i G).
D
-
ˆo
`
thi
.
con bˆo
.
phˆa
.
n
X´et d¯ˆo
`
thi
.
G = (V, E) v`a A ⊂ V, U ⊂ E. D
-
ˆo
`
thi
.
con bˆo
.
phˆa
.
n sinh bo
.

˙’
i tˆa
.
p A v`a U l`a d¯ˆo
`
thi
.
bˆo
.
phˆa
.
n cu
˙’
a G
A
sinh bo
.
˙’
i U.
1.2 Ma trˆa
.
n biˆe
˙’
u diˆe
˜
n d¯ˆo
`
thi
.
1.2.1 Ma trˆa

.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung
Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung cu
˙’
a d¯ˆo
`
thi
.
G = (V, E) l`a ma trˆa
.
n A = (a
ij
), i = 1, 2, . . . , n, j =
1, 2, . . . , m, v´o
.
i c´ac phˆa
`
n tu
.
˙’

0, 1 v`a −1, trong d¯´o mˆo
˜
i cˆo
.
t biˆe
˙’
u diˆe
˜
n mˆo
.
t cung cu
˙’
a G v`a mˆo
˜
i
h`ang biˆe
˙’
u diˆe
˜
n mˆo
.
t d¯ı
˙’
nh cu
˙’
a G. Nˆe
´
u e
k
= (v

i
, v
j
) ∈ E th`ı tˆa
´
t ca
˙’
c´ac phˆa
`
n tu
.
˙’
cu
˙’
a cˆo
.
t k bˇa
`
ng
khˆong ngoa
.
i tr`u
.
a
ik
= 1, a
jk
= −1.
13

V´ı du
.
1.2.1 Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung cu
˙’
a d¯ˆo
`
thi
.
trong H`ınh 1.3 l`a





e
1
e
2
e
3
e
4
e
5

a +1 +1 0 0 0
b −1 0 +1 +1 0
c 0 −1 −1 0 +1
d 0 0 0 −1 −1





.
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.................................................................................................................................................................

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a
b
d
c
• •
• •
e

1
e
2
e
3
e
4
e
5
H`ınh 1.3:
Nhˇa
´
c la
.
i rˇa
`
ng, ma trˆa
.
n vuˆong go
.
i l`a unimodular nˆe
´
u d¯i
.
nh th´u
.
c cu
˙’
a n´o bˇa
`

ng 1 hoˇa
.
c
−1. Ma trˆa
.
n A cˆa
´
p m × n go
.
i l`a total unimodular nˆe
´
u tˆa
´
t ca
˙’
c´ac ma trˆa
.
n vuˆong con khˆong
suy biˆe
´
n cu
˙’
a A l`a unimodular.
Mˆe
.
nh d¯ˆe
`
1.2.2 Ma trˆa
.
n liˆen thuˆo

.
c d¯ı
˙’
nh-cung cu
˙’
a d¯ˆo
`
thi
.
G = (V, E) l`a total unimodular.
Ch´u
.
ng minh. Ch´u ´y rˇa
`
ng ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung cu
˙’
a d¯ˆo
`
thi
.
G = (V, E) ch´u
.
a d¯´ung
hai phˆa

`
n tu
.
˙’
kh´ac khˆong trˆen mˆo
˜
i cˆo
.
t, mˆo
.
t bˇa
`
ng 1 v`a mˆo
.
t bˇa
`
ng −1. Do d¯´o ta c´o thˆe
˙’
ch´u
.
ng
minh theo quy na
.
p nhu
.
sau: Hiˆe
˙’
n nhiˆen, tˆa
´
t ca

˙’
c´ac ma trˆa
.
n vuˆong con khˆong suy biˆe
´
n cˆa
´
p 1
cu
˙’
a A l`a modular; gia
˙’
su
.
˙’
khˇa
˙’
ng d¯i
.
nh d¯´ung cho mo
.
i ma trˆa
.
n con khˆong suy biˆe
´
n cˆa
´
p (k − 1).
X´et ma trˆa
.

n vuˆong con A

cˆa
´
p k cu
˙’
a A. Nˆe
´
u mˆo
˜
i cˆo
.
t cu
˙’
a A

ch´u
.
a d¯´ung hai phˆa
`
n tu
.
˙’
kh´ac khˆong th`ı det(A

) = 0 (thˆa
.
t vˆa
.
y, tˆo

˙’
ng tˆa
´
t ca
˙’
c´ac h`ang cu
˙’
a A

l`a vector khˆong, do d¯´o c´ac
h`ang l`a d¯ˆo
.
c lˆa
.
p tuyˆe
´
n t´ınh). Nˆe
´
u tˆo
`
n ta
.
i mˆo
.
t cˆo
.
t cu
˙’
a A


khˆong c´o phˆa
`
n tu
.
˙’
kh´ac khˆong th`ı
det(A

) = 0. Cuˆo
´
i c`ung, nˆe
´
u tˆo
`
n ta
.
i cˆo
.
t j cu
˙’
a A

sao cho c´o d¯´ung mˆo
.
t phˆa
`
n tu
.
˙’
kh´ac khˆong

a
ij
(bˇa
`
ng 1, hay −1) th`ı det(A

) = ± det(A

), trong d¯´o A

l`a ma trˆa
.
n vuˆong cˆa
´
p (k − 1)
nhˆa
.
n d¯u
.
o
.
.
c t`u
.
A

bˇa
`
ng c´ach xo´a h`ang i v`a cˆo
.

t j. Theo gia
˙’
thiˆe
´
t quy na
.
p, det(A

) bˇa
`
ng 1, −1
hay 0 v`a do d¯´o mˆe
.
nh d¯ˆe
`
d¯u
.
o
.
.
c ch´u
.
ng minh. 
14

1.2.2 Ma trˆa
.
n liˆen thuˆo
.
c d¯ı

˙’
nh-ca
.
nh
X´et d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng G = (V, E). Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-ca
.
nh cu
˙’
a d¯ˆo
`
thi
.
G l`a ma trˆa
.
n
A = (a

ij
), i = 1, 2, . . . , n, j = 1, 2, . . . , m, v´o
.
i c´ac phˆa
`
n tu
.
˙’
0 v`a 1, trong d¯´o mˆo
˜
i cˆo
.
t biˆe
˙’
u diˆe
˜
n
mˆo
.
t ca
.
nh cu
˙’
a G v`a mˆo
˜
i h`ang biˆe
˙’
u diˆe
˜
n mˆo

.
t d¯ı
˙’
nh cu
˙’
a G; ngo`ai ra, nˆe
´
u ca
.
nh e
k
liˆen thuˆo
.
c
hai d¯ı
˙’
nh v
i
v`a v
j
th`ı tˆa
´
t ca
˙’
c´ac phˆa
`
n tu
.
˙’
cu

˙’
a cˆo
.
t k bˇa
`
ng khˆong ngoa
.
i tr`u
.
a
ik
= 1, a
jk
= 1.
V´ı du
.
1.2.3 Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-ca
.
nh cu
˙’
a d¯ˆo
`
thi
.

trong H`ınh 1.4 l`a





e
1
e
2
e
3
e
4
e
5
a 1 1 0 0 0
b 1 0 1 1 0
c 0 1 1 0 1
d 0 0 0 1 1





..................................................................................................................................................................................................................................................................................................................................
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...................................................................................................................................................................................................................................................................................................................................
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.
.
a
b
d
c
• •
• •
e
1
e
2
e
3
e

4
e
5
H`ınh 1.4:
Tr´ai v´o
.
i ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung, n´oi chung ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-ca
.
nh khˆong
total unimodular. Chˇa
˙’
ng ha
.
n, trong v´ı du
.
trˆen, ma trˆa
.
n con




1 1 0
1 0 1
0 1 1



c´o d¯i
.
nh th´u
.
c bˇa
`
ng −2.
D
-
ˆo
`
thi
.
vˆo hu
.
´o
.
ng G = (V, E) go
.
i l`a hai phˆa
`

n nˆe
´
u c´o thˆe
˙’
phˆan hoa
.
ch tˆa
.
p c´ac d¯ı
˙’
nh
V th`anh hai tˆa
.
p con r`o
.
i nhau V
1
v`a V
2
sao cho d¯ˆo
´
i v´o
.
i mˆo
˜
i ca
.
nh (v
i
, v

j
) ∈ E th`ı hoˇa
.
c
v
i
∈ V
1
, v
j
∈ V
2
hoˇa
.
c v
j
∈ V
1
, v
i
∈ V
2
.
15

.
.
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.
.

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.
.
.
a
b
c
d
e
• •
• • •
H`ınh 1.5: D
-
ˆo
`
thi
.
hai phˆa
`
n K
2,3
.
V´ı du
.
1.2.4 Dˆe
˜
kiˆe
˙’
m tra d¯ˆo
`
thi

.
K
2,3
trong H`ınh 1.5 l`a hai phˆa
`
n.
Mˆe
.
nh d¯ˆe
`
1.2.5 Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-ca
.
nh cu
˙’
a d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng G = (V, E) l`a total
unimodular nˆe

´
u v`a chı
˙’
nˆe
´
u G l`a d¯ˆo
`
thi
.
hai phˆa
`
n.
Ch´u
.
ng minh. (1) Nˆe
´
u d¯ˆo
`
thi
.
l`a hai phˆa
`
n, th`ı ch´ung ta c´o thˆe
˙’
ch´u
.
ng minh theo quy na
.
p rˇa
`

ng
mo
.
i ma trˆa
.
n vuˆong con B cu
˙’
a ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-ca
.
nh c´o d¯i
.
nh th´u
.
c det(B) = 0, 1 hoˇa
.
c
−1. D
-
iˆe
`
u n`ay d¯´ung v´o
.
i c´ac ma trˆa
.

n vuˆong con cˆa
´
p 1; gia
˙’
su
.
˙’
khˇa
˙’
ng d¯i
.
nh d¯´ung v´o
.
i c´ac ma
trˆa
.
n vuˆong con cˆa
´
p (k − 1). X´et ma trˆa
.
n vuˆong con B cˆa
´
p k.
Nˆe
´
u mˆo
˜
i cˆo
.
t B

j
cu
˙’
a B ch´u
.
a d¯´ung hai phˆa
`
n tu
.
˙’
bˇa
`
ng 1 th`ı

i∈I
1
B
i
=

i∈I
2
B
i
,
trong d¯´o I
1
v`a I
2
l`a c´ac tˆa

.
p chı
˙’
sˆo
´
tu
.
o
.
ng ´u
.
ng hai phˆan hoa
.
ch cu
˙’
a tˆa
.
p c´ac d¯ı
˙’
nh V v`a B
i
l`a
vector h`ang cu
˙’
a B. C´ac vector h`ang phu
.
thuˆo
.
c tuyˆe
´

n t´ınh, nˆen det(B) = 0.
Nˆe
´
u, ngu
.
o
.
.
c la
.
i, tˆo
`
n ta
.
i cˆo
.
t c´o d¯´ung mˆo
.
t phˆa
`
n tu
.
˙’
bˇa
`
ng 1, chˇa
˙’
ng ha
.
n b

ij
= 1, k´y hiˆe
.
u C
l`a ma trˆa
.
n nhˆa
.
n d¯u
.
o
.
.
c t`u
.
B bˇa
`
ng c´ach xo´a h`ang i v`a cˆo
.
t j. Th`ı
det(B) = ± det(C) (= 0, 1 hoˇa
.
c − 1 theo quy na
.
p).
(2) Mˇa
.
t kh´ac, dˆe
˜
d`ang ch´u

.
ng minh rˇa
`
ng ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-ca
.
nh cu
˙’
a d¯ˆo
`
thi
.
l`a mˆo
.
t chu
tr`ınh d¯ˆo
.
d`ai le
˙’
(t´u
.
c l`a sˆo
´
ca
.

nh trˆen chu tr`ınh l`a le
˙’
-xem Phˆa
`
n 1.3) c´o d¯i
.
nh th´u
.
c bˇa
`
ng ±2. Do
d¯´o G khˆong ch´u
.
a chu tr`ınh d¯ˆo
.
d`ai le
˙’
v`a v`ı vˆa
.
y n´o l`a hai phˆa
`
n theo bˆo
˙’
d¯ˆe
`
sau. 
Bˆo
˙’
d¯ˆe
`

1.2.6 D
-
ˆo
`
thi
.
vˆo hu
.
´o
.
ng G l`a hai phˆa
`
n nˆe
´
u v`a chı
˙’
nˆe
´
u G khˆong ch´u
.
a chu tr`ınh c´o d¯ˆo
.
d`ai le
˙’
.
Ch´u
.
ng minh. D
-
iˆe

`
u kiˆe
.
n cˆa
`
n. Do V d¯u
.
o
.
.
c phˆan hoa
.
ch th`anh V
1
v`a V
2
:
V = V
2
∪ V
2
, V
1
∩ V
2
= ∅.
16

Gia
˙’

thiˆe
´
t tˆo
`
n ta
.
i mˆo
.
t chu tr`ınh c´o d¯ˆo
.
d`ai le
˙’
µ = {v
i
1
, v
i
2
, . . . , v
i
q
, v
i
1
}
v`a khˆong mˆa
´
t t´ınh tˆo
˙’
ng qu´at, lˆa

´
y v
i
1
∈ V
1
. Do G l`a hai phˆa
`
n, nˆen hai d¯ı
˙’
nh liˆen tiˆe
´
p trˆen
chu tr`ınh µ pha
˙’
i c´o mˆo
.
t d¯ı
˙’
nh thuˆo
.
c V
1
v`a d¯ı
˙’
nh kia thuˆo
.
c V
2
. Do d¯´o v

i
2
∈ V
2
, v
i
3
∈ V
1
, . . . ,
v`a tˆo
˙’
ng qu´at, v
i
k
∈ V
1
nˆe
´
u k le
˙’
v`a v
i
k
∈ V
2
nˆe
´
u k chˇa
˜

n. M`a chu tr`ınh µ c´o d¯ˆo
.
d`ai le
˙’
nˆen
v
i
q
∈ V
1
v`a bo
.
˙’
i vˆa
.
y v
i
1
∈ V
2
. D
-
iˆe
`
u n`ay mˆau thuˆa
˜
n v´o
.
i V
1

∩ V
2
= ∅.
D
-
iˆe
`
u kiˆe
.
n d¯u
˙’
. Khˆong mˆa
´
t t´ınh tˆo
˙’
ng qu´at gia
˙’
thiˆe
´
t d¯ˆo
`
thi
.
G liˆen thˆong. Gia
˙’
su
.
˙’
khˆong tˆo
`

n
ta
.
i chu tr`ınh c´o d¯ˆo
.
d`ai le
˙’
.
Cho
.
n d¯ı
˙’
nh bˆa
´
t k`y, chˇa
˙’
ng ha
.
n v
i
v`a g´an nh˜an cho n´o l`a “ + ”. Sau d¯´o lˇa
.
p la
.
i c´ac ph´ep
to´an sau:
Cho
.
n d¯ı
˙’

nh d¯˜a d¯u
.
o
.
.
c g´an nh˜an v
j
v`a g´an nh˜an ngu
.
o
.
.
c v´o
.
i nh˜an cu
˙’
a v
j
cho tˆa
´
t ca
˙’
c´ac
d¯ı
˙’
nh kˆe
`
v´o
.
i d¯ı

˙’
nh v
j
.
Tiˆe
´
p tu
.
c qu´a tr`ınh n`ay cho d¯ˆe
´
n khi xa
˙’
y ra mˆo
.
t trong hai tru
.
`o
.
ng ho
.
.
p:
(a) Tˆa
´
t ca
˙’
c´ac d¯ı
˙’
nh d¯˜a d¯u
.

o
.
.
c g´an nh˜an v`a hai d¯ı
˙’
nh bˆa
´
t k`y kˆe
`
nhau c´o nh˜an kh´ac nhau (mˆo
.
t
mang dˆa
´
u + v`a mˆo
.
t mang dˆa
´
u −); hoˇa
.
c
(b) Tˆo
`
n ta
.
i d¯ı
˙’
nh, chˇa
˙’
ng ha

.
n v
j
k
, d¯u
.
o
.
.
c g´an hai nh˜an kh´ac nhau.
Trong Tru
.
`o
.
ng ho
.
.
p (a), d¯ˇa
.
t V
1
l`a tˆa
.
p tˆa
´
t ca
˙’
c´ac d¯ı
˙’
nh d¯u

.
o
.
.
c g´an nh˜an “+” v`a V
2
l`a tˆa
.
p tˆa
´
t
ca
˙’
c´ac d¯ı
˙’
nh d¯u
.
o
.
.
c g´an nh˜an “−”. Do tˆa
´
t ca
˙’
c´ac ca
.
nh liˆen thuˆo
.
c gi˜u
.

a c´ac cˇa
.
p d¯ı
˙’
nh c´o nh˜an
kh´ac nhau nˆen d¯ˆo
`
thi
.
G l`a hai phˆa
`
n.
Trong Tru
.
`o
.
ng ho
.
.
p (b), d¯ı
˙’
nh v
j
k
d¯u
.
o
.
.
c g´an nh˜an “+” do

.
c theo mˆo
.
t dˆay chuyˆe
`
n µ
1
n`ao
d¯´o, v´o
.
i c´ac d¯ı
˙’
nh d¯u
.
o
.
.
c g´an nh˜an “+” v`a “−” xen k˜e nhau xuˆa
´
t ph´at t`u
.
v
i
v`a kˆe
´
t th´uc ta
.
i
v
j

k
. Tu
.
o
.
ng tu
.
.
, d¯ı
˙’
nh v
j
k
d¯u
.
o
.
.
c g´an nh˜an “−” do
.
c theo mˆo
.
t dˆay chuyˆe
`
n µ
2
n`ao d¯´o, v´o
.
i c´ac
d¯ı

˙’
nh d¯u
.
o
.
.
c g´an nh˜an “+” v`a “−” xen k˜e nhau xuˆa
´
t ph´at t`u
.
v
i
v`a kˆe
´
t th´uc ta
.
i v
j
k
. Nhu
.
ng
nhu
.
thˆe
´
chu tr`ınh d¯i do
.
c theo µ
1

t`u
.
d¯ı
˙’
nh v
i
d¯ˆe
´
n d¯ı
˙’
nh v
j
k
sau d¯´o d¯i ngu
.
o
.
.
c la
.
i do
.
c theo µ
2
vˆe
`
la
.
i v
i

c´o d¯ˆo
.
d`ai le
˙’
. D
-
iˆe
`
u n`ay mˆau thuˆa
˜
n v´o
.
i gia
˙’
thiˆe
´
t, v`a do d¯´o khˆong thˆe
˙’
xa
˙’
y ra Tru
.
`o
.
ng
ho
.
.
p (b). D
-

i
.
nh l´y d¯u
.
o
.
.
c ch´u
.
ng minh. 
1.2.3 Ma trˆa
.
n kˆe
`
hay ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-d¯ı
˙’
nh
Gia
˙’
su
.
˙’
G = (V, E) l`a d¯ˆo
`

thi
.
sao cho c´o nhiˆe
`
u nhˆa
´
t mˆo
.
t cung liˆen thuˆo
.
c hai d¯ı
˙’
nh bˆa
´
t k`y v
i
v`a v
j
. Ma trˆa
.
n kˆe
`
hay ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-d¯ı
˙’

nh l`a ma trˆa
.
n vuˆong A = (a
ij
) cˆa
´
p n × n
17

v´o
.
i c´ac phˆa
`
n tu
.
˙’
0 hoˇa
.
c 1:
a
ij
:=

1 nˆe
´
u (v
i
, v
j
) ∈ E,

0 nˆe
´
u ngu
.
o
.
.
c la
.
i.
Trong tru
.
`o
.
ng ho
.
.
p d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng, ma trˆa
.
n kˆe
`
cu

˙’
a d¯o
.
n d¯ˆo
`
thi
.
c˜ung c´o thˆe
˙’
d¯u
.
o
.
.
c d¯i
.
nh
ngh˜ıa bˇa
`
ng c´ach xem mˆo
˜
i ca
.
nh (v
i
, v
j
) tu
.
o

.
ng ´u
.
ng hai cung (v
i
, v
j
) v`a (v
j
, v
i
). Trong tru
.
`o
.
ng
ho
.
.
p n`ay, ma trˆa
.
n kˆe
`
l`a d¯ˆo
´
i x´u
.
ng.
1.2.4 C´ac biˆe
˙’

u diˆe
˜
n cu
˙’
a d¯ˆo
`
thi
.
D
-
ˆe
˙’
mˆo ta
˙’
mˆo
.
t d¯ˆo
`
thi
.
, ta c´o thˆe
˙’
su
.
˙’
du
.
ng mˆo
.
t sˆo

´
c´ach biˆe
˙’
u diˆe
˜
n kh´ac nhau. V´o
.
i quan d¯iˆe
˙’
m
lˆa
.
p tr`ınh, n´oi chung c´ac biˆe
˙’
u diˆe
˜
n n`ay khˆong tu
.
o
.
ng d¯u
.
o
.
ng theo kh´ıa ca
.
nh hiˆe
.
u qua
˙’

cu
˙’
a
thuˆa
.
t to´an.
C´o hai c´ach biˆe
˙’
u diˆe
˜
n ch´ınh: Th´u
.
nhˆa
´
t, su
.
˙’
du
.
ng ma trˆa
.
n kˆe
`
hoˇa
.
c c´ac dˆa
˜
n xuˆa
´
t cu

˙’
a
n´o; th´u
.
hai, su
.
˙’
du
.
ng ma trˆa
.
n liˆen thuˆo
.
c hoˇa
.
c c´ac dˆa
˜
n xuˆa
´
t cu
˙’
a n´o.
Su
.
˙’
du
.
ng ma trˆa
.
n kˆe

`
Ch´ung ta biˆe
´
t rˇa
`
ng c´ac ma trˆa
.
n kˆe
`
cho ph´ep miˆeu ta
˙’
hoˇa
.
c c´ac 1-d¯ˆo
`
thi
.
d¯i
.
nh hu
.
´o
.
ng, hoˇa
.
c
c´ac d¯o
.
n d¯ˆo
`

thi
.
vˆo hu
.
´o
.
ng. V´o
.
i c´ach biˆe
˙’
u diˆe
˜
n d¯ˆo
`
thi
.
qua ma trˆa
.
n kˆe
`
, ta thˆa
´
y sˆo
´
lu
.
o
.
.
ng thˆong

tin, gˆo
`
m c´ac bit 0 v`a 1, cˆa
`
n lu
.
u tr˜u
.
l`a n
2
. C´ac bit c´o thˆe
˙’
d¯u
.
o
.
.
c kˆe
´
t ho
.
.
p trong c´ac t`u
.
. K´y
hiˆe
.
u w l`a d¯ˆo
.
d`ai cu

˙’
a t`u
.
(t´u
.
c l`a sˆo
´
c´ac bit trong mˆo
.
t t`u
.
m´ay t´ınh). Khi d¯´o mˆo
˜
i h`ang cu
˙’
a ma
trˆa
.
n kˆe
`
c´o thˆe
˙’
d¯u
.
o
.
.
c viˆe
´
t nhu

.
mˆo
.
t d˜ay n bit trong n/w t`u
.
1
. Do d¯´o sˆo
´
c´ac t`u
.
d¯ˆe
˙’
lu
.
u tr˜u
.
ma trˆa
.
n kˆe
`
l`a nn/w.
Ma trˆa
.
n kˆe
`
cu
˙’
a d¯ˆo
`
thi

.
vˆo hu
.
´o
.
ng l`a d¯ˆo
´
i x´u
.
ng, nˆen ta chı
˙’
cˆa
`
n lu
.
u tr˜u
.
nu
.
˙’
a tam gi´ac trˆen
cu
˙’
a n´o, v`a do d¯´o chı
˙’
cˆa
`
n n(n − 1)/2 bit. Tuy nhiˆen, v´o
.
i c´ach lu

.
u tr˜u
.
n`ay, s˜e tˇang d¯ˆo
.
ph´u
.
c
ta
.
p v`a th`o
.
i gian t´ınh to´an trong mˆo
.
t sˆo
´
b`ai to´an.
Trong tru
.
`o
.
ng ho
.
.
p c´ac ma trˆa
.
n thu
.
a (m  n
2

v´o
.
i d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng; m 
1
2
n(n + 1) d¯ˆo
´
i
v´o
.
i d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng) c´ach biˆe
˜
u diˆe
˜

n n`ay l`a l˜ang ph´ı. Do d¯´o ta s˜e t`ım c´ach biˆe
˙’
u diˆe
˜
n chı
˙’
c´ac
phˆa
`
n tu
.
˙’
kh´ac khˆong.
V`ı mu
.
c d¯´ıch n`ay ta s˜e su
.
˙’
du
.
ng mˆo
.
t ma
˙’
ng danh s´ach kˆe
`
cho d¯ˆo
`
thi
.

c´o hu
.
´o
.
ng. D
-
ˆo
`
thi
.
c´o
hu
.
´o
.
ng d¯u
.
o
.
.
c biˆe
˙’
u diˆe
˜
n bo
.
˙’
i mˆo
.
t ma

˙’
ng c´ac con tro
˙’
V out[1], V out[2], . . . , V out[n], trong d¯´o
mˆo
˜
i con tro
˙’
tu
.
o
.
ng ´u
.
ng v´o
.
i mˆo
.
t d¯ı
˙’
nh trong d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng. Mˆo
˜

i phˆa
`
n tu
.
˙’
cu
˙’
a ma
˙’
ng V out[i]
chı
˙’
d¯ˆe
´
n mˆo
.
t n´ut d¯ˆa
`
u lu
.
u tr˜u
.
mu
.
c d˜u
.
liˆe
.
u cu
˙’

a n´ut tu
.
o
.
ng ´u
.
ng d¯ı
˙’
nh v
i
v`a ch´u
.
a mˆo
.
t con tro
˙’
1
K´y hiˆe
.
u x l`a sˆo
´
nguyˆen nho
˙’
nhˆa
´
t khˆong b´e ho
.
n x.
18

chı
˙’
d¯ˆe
´
n mˆo
.
t danh s´ach liˆen kˆe
´
t cu
˙’
a c´ac d¯ı
˙’
nh kˆe
`
(d¯ı
˙’
nh d¯u
.
o
.
.
c nˆo
´
i v´o
.
i v
i
theo hu
.
´o

.
ng t`u
.
v
i
ra).
Mˆo
˜
i n´ut kˆe
`
c´o hai tru
.
`o
.
ng:
1. Tru
.
`o
.
ng sˆo
´
nguyˆen: lu
.
u tr˜u
.
sˆo
´
hiˆe
.
u cu

˙’
a d¯ı
˙’
nh kˆe
`
; v`a
2. Tru
.
`o
.
ng liˆen kˆe
´
t chı
˙’
d¯ˆe
´
n n´ut kˆe
´
tiˆe
´
p trong danh s´ach kˆe
`
n`ay.
.
.
.
.
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.

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v
1
v
2
v
3
v
4
v
5
v
6
• •
•
••
•

H`ınh 1.6:
C´ach biˆe
˙’
u diˆe
˜
n ma
˙’
ng danh s´ach kˆe
`
V out[] cu
˙’
a d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng trong H`ınh 1.6 d¯u
.
o
.
.
c
cho tu
.
o
.
ng ´u

.
ng trong H`ınh 1.7 (gia
˙’
su
.
˙’
c´ac mu
.
c d˜u
.
liˆe
.
u tu
.
o
.
ng ´u
.
ng c´ac d¯ı
˙’
nh theo th´u
.
tu
.
.
l`a
A, B, C, D, E, F ).
N´ut d¯ˆa
`
u

V out[1]
..........................................................
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•
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•
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A
v
4
v

5
NULL
V out[2]
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B
v

1
v
3
NULL
V out[3]
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C
v
3
NULL
V out[4]
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D
v
2
v
3
NULL
V out[5]
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E
v
3
v
6
NULL
V out[6]
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F
v
1
NULL

H`ınh 1.7: Danh s´ach kˆe
`
V out[] tu
.
o
.
ng ´u
.
ng d¯ˆo
`
thi
.
trong H`ınh 1.6.
Thay v`ı con tro
˙’
chı
˙’
d¯ˆe
´
n mˆo
.
t danh s´ach c´ac d¯ı
˙’
nh t`u
.
v
i
d¯i ra trong V out[i], ta tro
˙’
d¯ˆe

´
n
danh s´ach c´ac d¯ı
˙’
nh d¯i d¯ˆe
´
n v
i
v`a do d¯´o c´o thˆe
˙’
lu
.
u tr˜u
.
d¯ˆo
`
thi
.
thˆong qua ma
˙’
ng c´ac danh s´ach
19

kˆe
`
V in[i]. H`ınh 1.8 minh ho
.
a ma
˙’
ng c´ac danh s´ach kˆe

`
V in[] cu
˙’
a d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng trong H`ınh
1.6.
D
-
ˆe
˙’
´y rˇa
`
ng, c´ac sˆo
´
trong n´ut kˆe
`
cu
˙’
a V out[] (tu
.
o
.
ng ´u

.
ng, V in[]) l`a nh˜u
.
ng chı
˙’
sˆo
´
cˆo
.
t
(tu
.
o
.
ng ´u
.
ng, h`ang) trong ma trˆa
.
n kˆe
`
cu
˙’
a d¯ˆo
`
thi
.
m`a o
.
˙’
d¯´o sˆo

´
1 xuˆa
´
t hiˆe
.
n. Ngo`ai ra, trong
tru
.
`o
.
ng ho
.
.
p d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng, hai danh s´ach kˆe
`
n`ay l`a tr`ung nhau.
Khi d¯ˆo
`
thi
.
c´o tro
.

ng sˆo
´
, t´u
.
c l`a nˆe
´
u mˆo
˜
i cung hoˇa
.
c ca
.
nh e ∈ E c´o mˆo
.
t tro
.
ng lu
.
o
.
.
ng w(e),
ta chı
˙’
cˆa
`
n mo
.
˙’
rˆo

.
ng cˆa
´
u tr´uc cu
˙’
a mˆo
˜
i n´ut trong danh s´ach kˆe
`
bˇa
`
ng c´ach thˆem mˆo
.
t tru
.
`o
.
ng
lu
.
u tr˜u
.
tro
.
ng lu
.
o
.
.
ng cu

˙’
a cung.
C´ach biˆe
˙’
u diˆe
˜
n bˇa
`
ng danh s´ach kˆe
`
cu
˙’
a d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng c´o thˆe
˙’
d¯u
.
o
.
.
c c`ai d¯ˇa
.
t trong ngˆon

ng˜u
.
lˆa
.
p tr`ınh C v´o
.
i c´ac khai b´ao trong thu
.
viˆe
.
n Graph.h (xem Phu
.
lu
.
c A). D
-
ˆe
˙’
xˆay du
.
.
ng
ma
˙’
ng c´ac danh s´ach kˆe
`
V out[] v`a V in[] cho mˆo
.
t d¯ˆo
`

thi
.
, ta c´o thˆe
˙’
su
.
˙’
du
.
ng c´ac thu
˙’
tu
.
c
MakeV out() v`a MakeV in() tu
.
o
.
ng ´u
.
ng.
N´ut d¯ˆa
`
u
V in[1]
..........................................................
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A
v
2
v
6
NULL
V in[2]
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B
v
4
NULL
V in[3]
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.
C
v
2
v
3
v
4
v
5
NULL
V in[4]
..........................................................
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•
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•

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D
v

1
NULL
V in[5]
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•
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•
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.
E
v
1
NULL
V in[6]
..........................................................
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•
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•
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F
v
5
NULL
H`ınh 1.8: Danh s´ach kˆe
`
V in[] tu
.
o
.

ng ´u
.
ng d¯ˆo
`
thi
.
trong H`ınh 1.6.
Su
.
˙’
du
.
ng c´ac ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung hoˇa
.
c d¯ı
˙’
nh-ca
.
nh
Ma trˆa
.
n liˆen thuˆo
.
c d¯ı

˙’
nh-cung hoˇa
.
c d¯ı
˙’
nh-ca
.
nh cho ph´ep ch´ung ta mˆo ta
˙’
d¯ˆa
`
y d¯u
˙’
cˆa
´
u tr´uc cu
˙’
a
mˆo
.
t d¯a d¯ˆo
`
thi
.
khˆong c´o khuyˆen. Tuy nhiˆen, do chı
˙’
c´o hai phˆa
`
n tu
.

˙’
kh´ac khˆong trong mˆo
˜
i
cˆo
.
t, nˆen c´o thˆe
˙’
biˆe
˙’
u diˆe
˜
n thˆong tin o
.
˙’
da
.
ng th´ıch ho
.
.
p ho
.
n.
Ch´ung ta d¯i
.
nh ngh˜ıa hai ma
˙’
ng tuyˆe
´
n t´ınh α[] v`a β[] chiˆe

`
u m trong d¯´o v´o
.
i mˆo
˜
i cung
hoˇa
.
c ca
.
nh e
k
, k = 1, 2, . . . , m, c´ac gi´a tri
.
α[k] v`a β[k] l`a c´ac chı
˙’
sˆo
´
cu
˙’
a c´ac d¯ı
˙’
nh m`a e
k
liˆen
20

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v
1
v
2
v
3
•
••
e
1
e
2
e
3
e
4
e
5
e
6
e
7
e
8
H`ınh 1.9:
thuˆo

.
c. Trong tru
.
`o
.
ng ho
.
.
p c´o hu
.
´o
.
ng, ch´ung ta quyˆe
´
t d¯i
.
nh α[k] l`a d¯ı
˙’
nh gˆo
´
c v`a β[k] l`a d¯ı
˙’
nh
ngo
.
n cu
˙’
a cung e
k
.

Ch´u ´y rˇa
`
ng, tr´ai v´o
.
i ma trˆa
.
n kˆe
`
, c´ach biˆe
˙’
u diˆe
˜
n n`ay c˜ung c´o thˆe
˙’
d¯ˇa
.
c tru
.
ng cho c´ac d¯a
d¯ˆo
`
thi
.
c´o khuyˆen.
Chˇa
˙’
ng ha
.
n, d¯a d¯ˆo
`

thi
.
cu
˙’
a H`ınh 1.9 trong d¯´o c´ac cung d¯u
.
o
.
.
c d¯´anh sˆo
´
, ta nhˆa
.
n d¯u
.
o
.
.
c
k 1 2 3 4 5 6 7 8
α 1 3 1 2 2 3 2 2
β 3 1 3 1 1 2 3 2
Trong tru
.
`o
.
ng ho
.
.
p d¯ˆo

`
thi
.
c´o tro
.
ng sˆo
´
, ta chı
˙’
cˆa
`
n thˆem mˆo
.
t ma
˙’
ng w[] k´ıch thu
.
´o
.
c m lu
.
u
tr˜u
.
tro
.
ng lu
.
o
.

.
ng cu
˙’
a mˆo
˜
i ca
.
nh hoˇa
.
c cung v´o
.
i tu
.
o
.
ng ´u
.
ng mˆo
.
t-mˆo
.
t c´ac ma
˙’
ng α[] v`a β[].
Vˆe
`
c´ach kh´ac biˆe
˙’
u diˆe
˜

n hiˆe
.
u qua
˙’
ho
.
n cu
˙’
a d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng su
.
˙’
du
.
ng danh s´ach c´ac ca
.
nh
xem [43].
Mˆo
´
i liˆen hˆe
.
gi˜u

.
a c´ac biˆe
˙’
u diˆe
˜
n
Dˆe
˜
d`ang thˆa
´
y rˇa
`
ng tˆo
`
n ta
.
i c´ac thuˆa
.
t to´an d¯a th´u
.
c d¯ˆe
˙’
chuyˆe
˙’
n d¯ˆo
˙’
i gi˜u
.
a c´ac kiˆe
˙’

u d˜u
.
liˆe
.
u trˆen
d¯ˆo
`
thi
.
. H`ınh 1.10 minh ho
.
a c´ac kha
˙’
nˇang c´o thˆe
˙’
c´o.
D
-
ˆe
˙’
chuyˆe
˙’
n d¯ˆo
˙’
i gi˜u
.
a c´ac kiˆe
˙’
u d˜u
.

liˆe
.
u, cˆa
`
n c´ac chu
.
o
.
ng tr`ınh thu
.
.
c hiˆe
.
n d¯iˆe
`
u n`ay (b`ai
tˆa
.
p). C´ac biˆe
˙’
u diˆe
˜
n n`ay c´o thˆe
˙’
ca
˙’
i biˆen cho ph`u ho
.
.
p v´o

.
i yˆeu cˆa
`
u. Chˇa
˙’
ng ha
.
n, d¯ˆo
`
thi
.
c´o
21

Ma trˆa
.
n kˆe
`
d¯u
.
o
.
.
c kˆe
´
t theo h`ang
Ma trˆa
.
n kˆe
`

da
.
ng tu
.
`o
.
ng minh
Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung
hoˇa
.
c ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-ca
.
nh
da
.
ng tu
.
`o

.
ng minh
Ma trˆa
.
n kˆe
`
d¯u
.
o
.
.
c kˆe
´
t theo cˆo
.
t
Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung
hoˇa
.
c ma trˆa
.
n liˆen thuˆo
.
c d¯ı

˙’
nh-ca
.
nh
da
.
ng kˆe
´
t theo h`ang
Ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-cung
hoˇa
.
c ma trˆa
.
n liˆen thuˆo
.
c d¯ı
˙’
nh-ca
.
nh
da
.
ng kˆe

´
t theo cˆo
.
t
↑
|
O(n
2
)
|
↓
↑
|
O(mn)
|
↓
↑
|
O(mn)
|
↓
↑
|
O(m)
|
↓
↑
|
O(m)
|

↓
↑
|
O(m)
|
↓
H`ınh 1.10: Mˆo
´
i liˆen hˆe
.
v`a d¯ˆo
.
ph´u
.
c ta
.
p t´ınh to´an khi chuyˆe
˙’
n d¯ˆo
˙’
i gi˜u
.
a c´ac biˆe
˙’
u diˆe
˜
n kh´ac
nhau trong d¯ˆo
`
thi

.
.
tro
.
ng sˆo
´
c´o thˆe
˙’
d¯u
.
o
.
.
c biˆe
˙’
u diˆe
˜
n bo
.
˙’
i mˆo
.
t ma trˆa
.
n tro
.
ng lu
.
o
.

.
ng (c`on go
.
i l`a ma trˆa
.
n chi ph´ı
hay khoa
˙’
ng c´ach) cˆa
´
p n × n trong d¯´o phˆa
`
n tu
.
˙’
(i, j) bˇa
`
ng tro
.
ng lu
.
o
.
.
ng cu
˙’
a ca
.
nh hay cung
(v

i
, v
j
). Tuy nhiˆen, cˆa
`
n ch´u ´y rˇa
`
ng, t´ınh hiˆe
.
u qua
˙’
cu
˙’
a nhiˆe
`
u vˆa
´
n d¯ˆe
`
phu
.
thuˆo
.
c v`ao biˆe
˙’
u diˆe
˜
n
cu
˙’

a d¯ˆo
`
thi
.
. Do d¯´o, viˆe
.
c cho
.
n lu
.
.
a c´ac cˆa
´
u tr´uc d˜u
.
liˆe
.
u mˆo
.
t c´ach th´ıch ho
.
.
p l`a quan tro
.
ng.
22

1.3 T´ınh liˆen thˆong
1.3.1 Dˆay chuyˆe
`

n v`a chu tr`ınh
Gia
˙’
su
.
˙’
v
0
, v
k
l`a c´ac d¯ı
˙’
nh cu
˙’
a d¯ˆo
`
thi
.
vˆo hu
.
´o
.
ng G := (V, E). Dˆay chuyˆe
`
n µ t`u
.
v
0
d¯ˆe
´

n v
k
d¯ˆo
.
d`ai k l`a mˆo
.
t d˜ay xen k˜e (k + 1) d¯ı
˙’
nh v`a k ca
.
nh bˇa
´
t d¯ˆa
`
u t`u
.
v
0
v`a kˆe
´
t th´uc ta
.
i v
k
,
µ := {v
0
, e
1
, v

1
, e
2
, v
2
, . . . , v
k−1
, e
k
, v
k
},
trong d¯´o ca
.
nh e
i
liˆen thuˆo
.
c c´ac d¯ı
˙’
nh v
i−1
v`a v
i
, i = 1, 2, . . . , k. D
-
ˆe
˙’
gia
˙’

n tiˆe
.
n, ta thu
.
`o
.
ng viˆe
´
t
µ := {e
1
, e
2
, . . . , e
k
}.
Dˆay chuyˆe
`
n d¯u
.
o
.
.
c go
.
i l`a d¯o
.
n gia
˙’
n (tu

.
o
.
ng ´u
.
ng, so
.
cˆa
´
p) nˆe
´
u n´o khˆong d¯i hai lˆa
`
n qua
c`ung mˆo
.
t ca
.
nh (tu
.
o
.
ng ´u
.
ng, d¯ı
˙’
nh).
Chu tr`ınh l`a mˆo
.
t dˆay chuyˆe

`
n trong d¯´o d¯ı
˙’
nh d¯ˆa
`
u tr`ung v´o
.
i d¯ı
˙’
nh cuˆo
´
i. Chu tr`ınh qua
mˆo
˜
i ca
.
nh d¯´ung mˆo
.
t lˆa
`
n go
.
i l`a d¯o
.
n gia
˙’
n. Chu tr`ınh l`a so
.
cˆa
´

p nˆe
´
u n´o d¯i qua mˆo
˜
i d¯ı
˙’
nh d¯´ung
mˆo
.
t lˆa
`
n tr`u
.
d¯ı
˙’
nh d¯ˆa
`
u tiˆen hai lˆa
`
n (mˆo
.
t lˆa
`
n l´uc xuˆa
´
t ph´at v`a mˆo
.
t l´uc tro
.
˙’

vˆe
`
).
D
-
ˆo
`
thi
.
trong H`ınh 1.11 c´o
(a, e
1
, b, e
2
, c, e
3
, d, e
4
, b)
l`a dˆay chuyˆe
`
n t`u
.
d¯ı
˙’
nh a d¯ˆe
´
n d¯ı
˙’
nh b c´o d¯ˆo

.
d`ai bˆo
´
n. C´ac chu tr`ınh sau l`a so
.
cˆa
´
p
(b, e
2
, c, e
3
, d, e
4
, b), v`a (b, e
5
, f, e
7
, e, e
6
, b).
....................................................................................................................................................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................................................................................................................................................
.
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a
b
c
d
e
f
g
•• •
•

•• •
e
1
e
2
e
3
e
4
e
5
e
6
e
7
e
8
H`ınh 1.11:
Trong tru
.
`o
.
ng ho
.
.
p d¯ˆo
`
thi
.
khˆong c´o ca

.
nh song song (t´u
.
c l`a hai d¯ı
˙’
nh c´o nhiˆe
`
u nhˆa
´
t mˆo
.
t
ca
.
nh liˆen thuˆo
.
c ch´ung), d¯ˆe
˙’
d¯o
.
n gia
˙’
n dˆay chuyˆe
`
n µ d¯u
.
o
.
.
c viˆe

´
t la
.
i
µ = {v
0
, v
1
, v
2
, . . . , v
k
}.
23

1.3.2 D
-
u
.
`o
.
ng d¯i v`a ma
.
ch
Gia
˙’
su
.
˙’
v

0
, v
k
l`a c´ac d¯ı
˙’
nh cu
˙’
a d¯ˆo
`
thi
.
c´o hu
.
´o
.
ng G := (V, E). D
-
u
.
`o
.
ng d¯i µ t`u
.
v
0
d¯ˆe
´
n v
k
d¯ˆo

.
d`ai
k l`a mˆo
.
t d˜ay xen k˜e (k + 1) d¯ı
˙’
nh v`a k cung bˇa
´
t d¯ˆa
`
u t`u
.
v
0
v`a kˆe
´
t th´uc ta
.
i v
k
,
µ := {v
0
, e
1
, v
1
, e
2
, v

2
, . . . , v
k−1
, e
k
, v
k
},
trong d¯´o cung e
i
liˆen thuˆo
.
c c´ac d¯ı
˙’
nh v
i−1
v`a v
i
, i = 1, 2, . . . , k. D
-
ˆe
˙’
gia
˙’
n tiˆe
.
n, ta c´o thˆe
˙’
k´y
hiˆe

.
u d¯u
.
`o
.
ng d¯i µ l`a {e
1
, e
2
, . . . , e
k
}.
Do d¯´o trong H`ınh 1.12 d˜ay c´ac cung
µ
1
:= {e
6
, e
5
, e
9
, e
8
, e
4
}
µ
2
:= {e
1

, e
6
, e
5
, e
9
}
µ
3
:= {e
1
, e
6
, e
5
, e
9
, e
10
, e
6
, e
4
}
l`a c´ac d¯u
.
`o
.
ng d¯i.
D

-
u
.
`o
.
ng d¯i l`a d¯o
.
n gia
˙’
n nˆe
´
u khˆong ch´u
.
a cung n`ao qu´a mˆo
.
t lˆa
`
n. Suy ra c´ac d¯u
.
`o
.
ng d¯i
µ
1
, µ
2
l`a d¯o
.
n gia
˙’

n, nhu
.
ng d¯u
.
`o
.
ng d¯i µ
3
khˆong d¯o
.
n gia
˙’
n do n´o su
.
˙’
du
.
ng cung e
6
hai lˆa
`
n.
D
-
u
.
`o
.
ng d¯i l`a so
.

cˆa
´
p nˆe
´
u khˆong d¯i qua d¯ı
˙’
nh n`ao qu´a mˆo
.
t lˆa
`
n. Khi d¯´o d¯u
.
`o
.
ng d¯i µ
2
l`a
so
.
cˆa
´
p nhu
.
ng c´ac d¯u
.
`o
.
ng d¯i µ
1
v`a µ

3
l`a khˆong so
.
cˆa
´
p. Hiˆe
˙’
n nhiˆen, d¯u
.
`o
.
ng d¯i so
.
cˆa
´
p l`a d¯o
.
n
gia
˙’
n nhu
.
ng ngu
.
o
.
.
c la
.
i khˆong nhˆa

´
t thiˆe
´
t d¯´ung. Chˇa
˙’
ng ha
.
n, ch´u ´y rˇa
`
ng d¯u
.
`o
.
ng d¯i µ
1
l`a d¯o
.
n
gia
˙’
n nhu
.
ng khˆong so
.
cˆa
´
p, d¯u
.
`o
.

ng d¯i µ
2
v`u
.
a d¯o
.
n gia
˙’
n v`a v`u
.
a so
.
cˆa
´
p, d¯u
.
`o
.
ng d¯i µ
3
khˆong
d¯o
.
n gia
˙’
n c˜ung khˆong so
.
cˆa
´
p.

Ch´u ´y rˇa
`
ng, kh´ai niˆe
.
m dˆay chuyˆe
`
n l`a ba
˙’
n sao khˆong c´o hu
.
´o
.
ng cu
˙’
a d¯u
.
`o
.
ng d¯i v`a ´ap
du
.
ng cho c´ac d¯ˆo
`
thi
.
m`a khˆong d¯ˆe
˙’
´y d¯ˆe
´
n hu

.
´o
.
ng cu
˙’
a c´ac cung.
D
-
u
.
`o
.
ng d¯i c˜ung c´o thˆe
˙’
d¯u
.
o
.
.
c biˆe
˙’
u diˆe
˜
n bo
.
˙’
i d˜ay c´ac d¯ı
˙’
nh m`a ch´ung d¯i qua trong tru
.

`o
.
ng
ho
.
.
p khˆong c´o cung song song (t´u
.
c hai cung c´o c`ung gˆo
´
c v`a c`ung ngo
.
n). Do d¯´o, d¯u
.
`o
.
ng d¯i
µ
1
c´o thˆe
˙’
biˆe
˙’
u diˆe
˜
n bo
.
˙’
i d˜ay d¯ı
˙’

nh {v
2
, v
5
, v
4
, v
3
, v
5
, v
6
}.
Ma
.
ch l`a mˆo
.
t d¯u
.
`o
.
ng d¯i {e
1
, e
2
, . . . , e
k
} trong d¯´o d¯ı
˙’
nh gˆo

´
c cu
˙’
a cung e
1
tr`ung v´o
.
i d¯ı
˙’
nh
ngo
.
n cu
˙’
a cung e
k
. Do d¯´o d¯u
.
`o
.
ng d¯i {e
5
, e
9
, e
10
, e
6
} trong H`ınh 1.12 l`a ma
.

ch.
1.3.3 T´ınh liˆen thˆong
D
-
ˆo
`
thi
.
vˆo hu
.
´o
.
ng go
.
i l`a liˆen thˆong nˆe
´
u tˆa
´
t ca
˙’
c´ac cˇa
.
p d¯ı
˙’
nh v
i
v`a v
j
tˆo
`

n ta
.
i dˆay chuyˆe
`
n t`u
.
v
i
d¯ˆe
´
n v
j
. Quan hˆe
.
v
i
Rv
j
nˆe
´
u v`a chı
˙’
nˆe
´
u v
i
= v
j
hoˇa
.

c tˆo
`
n ta
.
i mˆo
.
t dˆay chuyˆe
`
n nˆo
´
i hai d¯ı
˙’
nh v
i
v`a v
j
l`a quan hˆe
.
tu
.
o
.
ng d¯u
.
o
.
ng (pha
˙’
n xa
.

, d¯ˆo
´
i x´u
.
ng v`a bˇa
´
c cˆa
`
u).
24

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e
7
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e
9
v
1
v
2
v
3
v
4
v
5
v
6
•
•
•
•
•

•
H`ınh 1.12:
L´o
.
p tu
.
o
.
ng d¯u
.
o
.
ng trˆen V x´ac d¯i
.
nh bo
.
˙’
i quan hˆe
.
tu
.
o
.
ng d¯u
.
o
.
ng R phˆan hoa
.
ch tˆa

.
p V
th`anh c´ac tˆa
.
p con r`o
.
i nhau V
1
, V
2
, . . . , V
p
. Sˆo
´
p go
.
i l`a sˆo
´
th`anh phˆa
`
n liˆen thˆong cu
˙’
a d¯ˆo
`
thi
.
.
Theo d¯i
.
nh ngh˜ıa, d¯ˆo

`
thi
.
liˆen thˆong nˆe
´
u v`a chı
˙’
nˆe
´
u sˆo
´
th`anh phˆa
`
n liˆen thˆong bˇa
`
ng mˆo
.
t. C´ac
d¯ˆo
`
thi
.
con G
1
, G
2
, . . . , G
p
sinh bo
.

˙’
i c´ac tˆa
.
p con V
1
, V
2
, . . . , V
p
go
.
i l`a c´ac th`anh phˆa
`
n liˆen thˆong
cu
˙’
a d¯ˆo
`
thi
.
. Mˆo
˜
i th`anh phˆa
`
n liˆen thˆong l`a mˆo
.
t d¯ˆo
`
thi
.

liˆen thˆong.
H`ınh 1.13 minh ho
.
a d¯ˆo
`
thi
.
c´o ba th`anh phˆa
`
n liˆen thˆong.
.
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.......................................................................................................................................................................................................................
v
8
v
3
v
6
•
• •
•
• •
•
•
H`ınh 1.13: D

-
ˆo
`
thi
.
c´o ba th`anh phˆa
`
n liˆen thˆong.
X´ac d¯i
.
nh sˆo
´
th`anh phˆa
`
n liˆen thˆong cu
˙’
a d¯ˆo
`
thi
.
l`a mˆo
.
t trong nh˜u
.
ng b`ai to´an co
.
ba
˙’
n cu
˙’

a
l´y thuyˆe
´
t d¯ˆo
`
thi
.
v`a c´o nhiˆe
`
u ´u
.
ng du
.
ng trong thu
.
.
c tiˆe
˜
n; chˇa
˙’
ng ha
.
n, x´ac d¯i
.
nh t´ınh liˆen thˆong
cu
˙’
a ma
.
ch d¯iˆe

.
n, ma
.
ng d¯iˆe
.
n thoa
.
i, v.v.
Ch´ung ta s˜e tr`ınh b`ay mˆo
.
t sˆo
´
thuˆa
.
t to´an c´o th`o
.
i gian O(m) gia
˙’
i b`ai to´an n`ay v`ı n´o
25

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