LTDT- Bai 02 - Tong Quan LTDT Sep19 2011 Slide
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Transcript of LTDT- Bai 02 - Tong Quan LTDT Sep19 2011 Slide
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15/09/2011
1
Bi ging L thuyt th(cp nht thng 01/2010)
ng Nguyn c Tin
th (graph) l mt cu trc ri rc gm cc nh v cc cnh ni cc nh .
th c k hiu l G = (V, E), trong :
V l tp nh (vertex),
E V V l tp hp cc cnh (edge).
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 2
CC LOI TH
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 3 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin Giang
Tr Vinh
Vnh Long
ng Thp
Cn Th
4
Mt n th (simple graph) G = (V, E) gm mt tp khng rng V v mt tp cnh E l cc cnh khng sp th t ca cc nh phn bit.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 5 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin Giang
Tr Vinh
Vnh Long
ng Thp
Cn Th
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Mt a th (multigraph) G = (V, E) gm mt tp cc nh V, mt tp cc cnh E v mt hm f t E ti {{u, v} | u, v V, u v}. Cc cnh e1, e2 c gi l cnh song song (parallel) (hay cnh bi (multiple)) nu f(e1) = f(e2).
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 7 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin Giang
Tr Vinh
Vnh Long
ng Thp
Cn Th
8
Mt gi th (pseudo graph) G = (V, E) gm mt tp nh V, mt tp cc cnh E v mt hm f t E ti {{u, v} | u, v V}.
Mt cnh l khuyn (loop) nu f(e) = {u, u} = {u} vi mt nh u no .
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
9 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin Giang
Tr Vinh
Vnh Long
ng Thp
Cn Th
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Mt th c hng (directed graph hoc digraph)G = (V, E) gm tp cc nh V v tp cc cnh E l cc cp c th t ca cc phn t thuc V. Cc cnh y cn c gi l cung (arc).
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 11 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin Giang
Tr Vinh
Vnh Long
ng Thp
Cn Th
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Mt a th c hng (directed multigraph)G = (V, E) gm mt tp cc nh V, tp cc cnh E v mt hm f t E ti {{u, v} | u, v V}.
Cc cnh e1 v e2 l cc cnh bi nu f(e1) = f(e2).
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 13
Loi Cnh C cnh bi? C khuyn?
n th V hng Khng Khng
a th V hng C Khng
Gi th V hng C C
th c hng C hng Khng C
a th c hng C hng C C
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 14
Ch lm vic vi th c tp nh v tp cnh hu hn. Thut ng th c ngm hiu l th hu hn.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 15
CC M HNH TH
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 16
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Gu trc
Diu hu
Sc Qu
C
Chim g kin
Chut
Chut ch
Th c ti
17 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin
Nam
Minh Vn
Th
c
th quen bit trn tri t c hn 6 t nh v c th hn t t cnh!
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HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Linda
Brian
RaffaAndy
John
Steve
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nh: tc gi
Cnh: 2 ngi vit chung mt bi bo
th cng tc (2001) c hn 337000 nh v 496200 cnh.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 20
1. A = 0
2. B = 1
3. C = A + 1
4. D = B + A
5. E = D + 1
6. E = C + D
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
S1 S2
S3 S4
S5S6
21
1. Xc nh loi th ca cc th sau:
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
a
b
c
d
22
2. Xy dng th nh hng cho cc thnh vin lnh o ca mt cng ty nu:
Ch tch c nh hng ln gim c nghin cu & pht trin, gim c marketing, gim c iu hnh;
Gim c nghin cu & pht trin c nh hng ln gim c iu hnh;
Gim c Marketing nh hng ln Gim c iu hnh;
Khng ai c th nh hng ln trng phng ti chnh v Trng phng ti chnh khng nh hng ln bt c ai.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 23
3. Hy xy dng th u tin cho chng trnh sau:
1. x = 0;
2. x = x + 1;
3. y = 2;
4. z = y;
5. x = x + 2;
6. y = x + z;
7. z = 4;
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 24
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CNH K, NH K, BC
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 25
Hai nh u v v trong mt th v hng G c gi l k hay lin k (adjacent) (hay lng ging (neibor)) nu {u, v} l mt cnh ca G.
Nu e = {u, v} th e c gi l cnh lin thuc (incident) vi cc nh u v v. Cnh e cng c gi l cnh ni (connect) cc nh u v v.
Cc nh u v v gi l cc im u mt (endpoint)ca cnh {u, v}.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 26
Bc (degree) ca mt nh trn th v hng l s cc cnh lin thuc vi n, ring khuyn ti mt nh c tnh hai ln cho bc ca n. Ngi ta k hiu bc ca nh v l deg(v).
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
27 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin Giang
Tr Vinh
Vnh Long
ng Thp
Cn Th
3
6
5
3
5
28
Cho G = (V, E) l mt th v hng c e cnh. Khi:
Chng minh:
C bao nhiu cnh trong th c 10 nh, mi nhc bc bng 7?
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Vv
ve )deg(2
29
nh l: Mt th v hng c s lng nh bcl l mt s chn.
Chng minh:
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 30
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Khi (u, v) l cnh ca th c hng G, th u c gi l ni ti v v v c gi l ni t u. nh u c gi l nh u (initial vertex), nh v gi l nh cui (terminal hoc end vertex) ca cnh (u, v).
Cnh e = (u, v) c gi l i t nh u ti nh v hoc i ra nh u vo nh v.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 31
Trong th c hng, bc vo (in-degree) ca nh v, k hiu l deg-(v) l s cc cnh c nh cui l v.
Bc ra (out-degree) ca nh v, k hiu l deg+(v) l s cc cnh c nh u l v.(Mt khuyn s gp thm 1 n v vo bc vo v 1 n v vo bc ra ca nh cha khuyn)
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 32
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin Giang
Tr Vinh
Vnh Long
ng Thp
Cn Th
d- = 2, d+ = 1
d- = 4, d+ = 1
d- = 2, d+ = 3
d- = 2, d+ = 4
d- = 1, d+ = 2
33
Cho G = (V, E) l mt th c hng. Khi :
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
VvVv
Evv )(deg)(deg
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nh treo (pendant vertex, end vertex) l nh c bcbng 1.
nh c lp (isolated vertex) l nh c bc bng 0.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 35 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin Giang
Tr Vinh
Vnh Long
ng Thp
Cn Th
Ph Quc
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MT S N TH C BIT
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 37
th y (complete graph) n nh, k hiu l Kn, l mt n th cha ng mt cnh ni mi cp nh phn bit.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 38
th chnh quy (regular graph) l n th m bc ca mi nh u bng nhau.
Nu bc ca cc nh l n, th th ny c gi l n chnh quy (n regular).
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 39 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
2 chnh quy
3 chnh quy
4 chnh quy
40
th vng (cycle) Cn, n 3 l mt th c n nh v1, v2 vn v cc cnh {v1, v2}, {v2, v3}, {vn-1, vn} v {vn, v1}.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 41
Mt n th G c gi l th phn i (bipartite graph) nu tp cc nh V c th phn lm hai tp con khng rng, ri nhau V1 v V2 sao cho mi cnh ca th ni mt nh ca V1 vi mt nh ca V2.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 42
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HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 43
th C6 v K3 c phi l cc th phn i? Gii thch.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
1 2
3
45
6
1
2
3
44
th phn i y (complete bipartite graph)Km,n l th c tp nh c phn thnh hai tp con tng ng c m nh v n nh v c mt cnh gia 2 nh nu v ch nu mt nh thuc tp con ny v nh th hai thuc tp con kia.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 45 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 46
th con (subgraph) ca th G = (V, E) l th H = (W, F) trong W V v V E.
th H l con ca th G c gi l th b phn (spanning subgraph) ca G khi W = V.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 47 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
1 2
3
45
6
1
3
45
6
1 2
3
45
6
48
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Th no l 2 nh k nhau?
Bc ca nh l g?
Mi lin quan gia s cnh v bc?
S lng nh bc l ca mt th?
nh treo? nh c lp?
th y ?
th vng?
th phn i?
th phn i y ?
th con? th b phn
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 49
1. Cho G l mt th n, v hng c s nh n > 3.
Chng minh G c cha 2 nh cng bc.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 50
2. C th tn ti th n c 15 nh, mi nh c bc
bng 5 hay khng?
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 51
3. Trong mt bui chiu i, mi ngi u bt tay
nhau. Chng t rng tng s ngi c bt tay vi
mt s l ngi khc l mt s chn. Gi s khng ai
t bt tay mnh
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 52
4. V cc th:
a. K7
b. K1,8
c. K4,4
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 53
5. Cc th sau y c bao nhiu cnh?
a. Kn
b. Km,n
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 54
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6. th s c bao nhiu cnh nu n c cc nh bc 4, 3, 3, 2,
2? V mt th nh vy.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 55
7. C tn ti th n cha nm nh vi cc bc sau
y? Nu c hy v th .
a. 3, 3, 3, 3, 2
b. 1, 2, 3, 4 ,5
c. 1, 1, 1, 1, 1
d. 1, 2, 1, 2, 1
e. 2, 1, 0, 2, 1
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 56
8. Mt bui tic c 6 ngi tham d. Chng minh rngc 3 ngi tng cp quen nhau hoc 3 ngikhng quen nhau.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 57
9. Cc th sau y c phi l th phn ikhng?
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 58
TH NG CU
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 59
Cc n th G1 = (V1, E1) v G2 = (V2, E2) l ngcu (isomorphic) nu c hm song nh f t V1 ln V2sao cho cc nh u v v l lin k trong G1 nu vch nu f(u) v f(v) l lin k trong G2 vi mi u, v trong V1. Hm f nh vy c gi l mt ng cu.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 60
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HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
th G1 v G2 l ng cu vi nhau:
f(1) = a, f(2) = b, f(3) = c, f(4) = d
e1 = E1, e2 = E2
1
3
2
4
a
b c
d
e1
e2
e3
e4e5
e6
E1 E2
E3
E4
E5
E6
61
xc nh xem mt th l ng cu hay khng l rt kh khn!
chng minh 2 th l ng cu, cn a ra mt quan h tng ng (ng cu) gia 2 th ny.
chng minh 2 th khng ng cu, ch ra chng khng c chung mt tnh cht m cc th ng cu phi c.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 62
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
ca
e d
b
zx
v t
y
63 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
ba
d c
e f
h g
ts
v u
w x
z y
64
BIU DIN TH
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 65
nh Cc nh k
a b, e
b a, c
c b, d, e
d c, e
e a, d
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
ca
e d
b
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nh u nh cui
1 2, 3, 4, 6
2 3
3
4 3
5
6 3
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
1 2
3
45
6
67
Gi s G = (V, E) trong V = {v1, v2, }, |V| = n.
Ma trn k (Adjacency Matrix) A (hay AG) ca G l mt ma trn 0-1 cp nxn c phn t aij ti dng i, ct j bng 1 nu vi v vj k nhau v bng 0 nu vi v vjkhng k nhau.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 68
a b c d e
a 0 1 0 0 1
b 1 0 1 0 0
c 0 1 0 1 1
d 0 0 1 0 1
e 1 0 1 1 0
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
ca
e d
b
69 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
1 2
3
45
6
1 2 3 4 5 6
1 0 1 1 1 0 1
2 0 0 1 0 0 0
3 0 0 0 0 0 0
4 0 0 1 0 0 0
5 0 0 0 0 0 0
6 0 0 1 0 0 0
70
1 2 3 4
1 1 1 0 1
2 1 1 2 0
3 0 2 1 2
4 1 0 2 0
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
21
4 3
71
Gi s G = (V, E) trong :
V = {v1, v2, }, |V| = n.
E = {e1, e2, }, |E| = e.
Ma trn lin thuc (incidence matrix) M ca G l mt ma trn 0-1 kch thc nxe c phn t aij ti dng i, ct j bng 1 nu cnh ej ni vi nh vi v bng 0 nu cnh ej khng ni vi nh vi.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 72
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1 2 3 4 5 6
a 1 0 1 0 0 0
b 1 1 0 0 0 0
c 0 1 0 1 0 1
d 0 0 0 1 1 0
e 0 0 1 0 1 1
(a, b) (b, c) (a, e) (c, d) (d, e) (c, e)
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
ca
e d
be1 e2
e3e4
e5
e6
73
1 2 3 4 5 6 7 8 9
1 1 1 0 0 0 0 0 0 1
2 0 1 1 1 1 0 0 0 0
3 0 0 0 1 1 1 1 1 0
4 0 0 0 0 0 0 1 1 1
(1) (1, 2) (2) (2, 3) (2, 3) (3) (3, 4) (3, 4) (1, 4)
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
21
4 3
e1
e2e3
e5e4
e6
e7
e8
e9
74
Danh sch k
Cch 1:
int ke[MAX][MAX];
int soDinhKe[MAX];
Cch 2:
list ke[MAX];
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 75
Ma trn k & Ma trn lin thuc
int a[MAX][MAX];
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 76
Chn la cch biu din no l ph hp?
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 77
Hai nh l sau y s gip chng ta xc nh s ng cu ca hai th ( gn nhn):
nh l 1: Hai th l ng cu vi nhau khi v ch khi cc nh ca n c th c gn nhn sao cho ma trn k tng ng l ging nhau.
nh l 2: Hai th c gn nhn G1 v G2 vi hai ma trn k tng ng l A1 v A2 ng cu vi nhau khi v ch khi tn ti ma trn hon v P sao cho
P A1 PT = A2
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Ghi ch: Ma trn hon v l mt ma trn c c bng cch hon v cc hng v/hoc ctca mt ma trn n v n x n. Nh vy ma trn hon v l mt ma trn vung m mi hng
v ct ch c 1 phn t c gi tr '1', cc phn t cn li c gi tr '0'.
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Xt hai th G1 v G2 c gn nhn nh bn di v hai ma trn k tng ng:
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
21
4 3
da
b c
1 2 3 4
1 0 1 0 1
2 1 0 1 0
3 0 1 0 1
4 1 0 1 0
a b c d
a 0 0 1 1
b 0 0 1 1
c 1 1 0 0
d 1 1 0 0
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Xt ma trn hon v
Ta c:
Vy G1 v G2 ng cu vi nhau theo nh x: 1 a, 2 c, 3 b v 4 d.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
1000
0010
0100
0001
P
21
0011
0011
1100
1100
1000
0010
0100
0001
0101
1010
0101
1010
1000
0010
0100
0001
APPA T
80
chng minh 2 th khng ng cu, ta ch cn a ra v d phn chng.
chng minh 2 th l ng cu, cn phi ch ra ma trn hon v P.
(C n n! hon v khc nhau vi th n nh, do vy, bi ton xc nh 2 th ng cu c phc tp rt ln!)
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 81
1. Hy biu din cc th sau y bng 3 cch biu din hc. Liu c th biu din c?
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
21
4 3
21
4 3
5
82
2. Hy biu din cc th sau bng cc cch biu din hc
a) K4
b) K1,4
c) C4
d) K2, 3
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 83
3. Hy v cc th v hng cho bi ma trn k sau:
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
1 3 2
3 0 4
2 4 0
1 2 0 1
2 0 3 0
0 3 1 1
1 0 1 0
0 1 3 0 4
1 2 1 3 0
3 1 1 0 1
0 3 0 0 2
4 0 1 2 3
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4. Hy m t hng v ct ca ma trn k ca th tng ng vi nh c lp.
5. Cc n th vi ma trn k sau y c ng cu khng?
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
0 0 1
0 0 1
1 1 0
0 1 1
1 0 0
1 0 0
0 1 0 1
1 0 0 1
0 0 0 1
1 1 1 0
0 1 1 1
1 0 0 1
1 0 0 1
1 1 1 0
85
6. Cc th sau y c ng cu?
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
a)
b)
86
7. Cc th sau y c ng cu?
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
a)
b)
87
NG I, CHU TRNH
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 88
ng i (path) ( di n) t u ti v trong mt th v hng l mt dy cc cnh e1, e2 en ca th sao cho f(e1) = {x0, x1}, f(e2) = {x1, x2} f(en) = {xn-1, xn}, vi x0 = u v xn = v.
Khi th l n ta k hiu ng i ny bng dy cc nh x0, x1, xn.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 89
ng i c gi l chu trnh (cycle/circuit) nu n v bt u v kt thc ti mt nh (ngha l u = v).
ng i hay chu trnh gi l n nu n khng i qua cng mt cnh qu mt ln.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 90
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{a, b, c, e, d} l mt ng i di 4.
{a, b, e, d} khng l ng i.
{a, b, c, e, a} l mt chu trnh.
{c, e, d, e, c} khng phi l mt ng i n.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
ca
e d
b
91
Nhiu nh khoa hc x hi cho rng hu ht mi cp ngi trn th gii c kt ni bi mt chui kh nh, c th ch gm 5 ngi hoc t hn.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
Tin
Nam
Minh Vnc
Th
92
Nu 2 th ng cu vi nhau, n s c cc chu trnh c cng di k vi nhau, trong k > 2.
(iu ny suy ra Nu iu kin trn khng tha, ngha l 2 th khng ng cu vi nhau)
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 93 HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
a
f b
e c
d
1
6 2
5 3
4a
f b
e c
1
5 2
4 3
94
LIN THNG
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 95
Mt th v hng c gi l lin thng (connected) nu c ng i gia mi cp nh phn bit ca th.
Ngc li, th ny c gi l khng lin thng (disconnected).
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 96
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HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 97
Mt th khng lin thng s bao gm nhiu th con lin thng, cc th con ny c gi l cc thnh phn lin thng (connected component).
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 98
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
a b
c
k l
g
d e
h
i j
f
99
nh l: Nu mt th G (khng quan tm linthng hay khng) c ng 2 nh bc l, chc chns c mt ng i ni 2 nh ny.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 100
S cnh ti a ca mt n th khng lin thng G gm n nh v k thnh phn l:
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
2
)1)(( knkn
101
th c hng gi l lin thng mnh (strongly connected) nu c ng i t a ti b v t b ti a vi mi cp nh a v b ca th.
th c hng gi l lin thng yu (weakly connected) nu c ng i gia 2 nh bt k ca th v hng nn (underlying graph).
th c hng c gi l lin thng mt phn(unilaterally connected) nu vi mi cp nh a, b bt k, c t nht mt nh n c nh cn li.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 102
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HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
1 2
3
4
5
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3
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nh khp (cut vertex/ articulation point) ca mt th v hng l nh m nu xa nh ny khi th v cc cnh ni n n th s thnh phn lin thng ca th s tng thm.
Cnh cu (bridge) ca mt th v hng l cnh m nu xa i khi th th s thnh phn lin thng ca th s tng thm.
th song lin thng (biconnectivity) l th khng cha nh khp.
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Kh nng lin thng (connectivity) hay kh nng lin thng nh (vertex connectivity) (G) ca mt th l tp c lc lng nh nht cc nh m khi xa cc nh ny i th khng cn lin thng na.
Mt th G c gi l k lin thng (kconnectivity) hay k lin thng nh (k-vertex-connectivity) khi (G) k.
Nh vy, mt th y Kn, theo nh ngha s c (G) = n 1.
th song lin thng chnh l th 2 lin thng.
Ghi ch: c l kappa.
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1. Mi danh sch cc nh sau y c to nn ng i trong th cho hay khng? ng i no l ng i n? ng i no l chu trnh? di ca cc ng i?a) a, e, b, c, d
b) a, e, a, d, b, c, a
c) e, b, a, d, b, e
d) c, b, d, a, e, c
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2. Cc thnh phn lin thng trong th quen bit biu din ci g?
3. Hy tm thnh phn lin thng mnh ca cc th c hng sau:
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4. Cc th sau y c phi l th song lin thng?
a) K3b) K4c) K2,3 d) C5
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 109
CC PHP DUYT TH
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 110
Depth First Search
Nguyn l: Khi t mt nh, i theo cc cnh (cung) xa nht c th. Tr li nh sau ca cnh xa nht, tip tc duyt nh trc cho n nh cui cng.
tng: nh thm cng mun s cng sm tr thnh nh duyt xong.
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procedure DFS(v)
begin
Thmnh(v);
chaXt[v] = false;
for u K(V) do
if chaXt[u] then
DFS(u);
end
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ChaXt[ ] a b c d e f g H
Khi to T T T T T T T T
Thm ln 1 F T T T T T T T
Thm ln 2 F F T T T T T T
Thm ln 3 F F F T T T T T
Thm ln 4 F F F T T T F T
Thm ln 5 F F F F T T F T
Thm ln 6 F F F F F T F T
Thm ln 7 F F F F F F F T
Thm ln 8 F F F F F F F F
F: nh hin tiF: nh chn c bc k tip.
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Chng trnh chnh
BEGIN
for v V do
chaXt[v] = true;
for v V do
if chaXt[v] then
DFS(v);//BFS(v)
END.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 115
Breadth First Search
tng: nh thm cng sm s cng sm thnh nh duyt xong.
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procedure BFS(v)
begin
QUEUE = ;
QUEUE v; // Kt np v vo QUEUEchaXt[v] = false;
while QUEUE do
p QUEUE;Thmnh(p);
for u K(p) do
if chaXt[u] then
QUEUE u;chaXt[u] = false;
end;
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HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin
ChaXt[ ] a b c d e f g H
Khi to T T T T T T T T
Thm ln 1 F T T T T T T T
Thm ln 2 F F T T T T T T
Thm ln 3 F F F T T T T T
Thm ln 4 F F F T T T F T
Thm ln 5 F F F F T T F T
Thm ln 6 F F F F T T F F
Thm ln 7 F F F F F T F F
Thm ln 8 F F F F F F F F
F: nh hin tiF: nh chn c bc k tip.
a b
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d e
h f
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5 7
86
119
V D MINH HA
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Ln cn 4 Ln cn 8
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Yu cu
Khi kho st cc o mt vng bin, ngi ta ghi kt qu kho st li thnh mt bn nh phn trong s 0 cho bit bin v s 1 l t lin. Mt o chnh l mt min lin thng 4 cc s 1 trn bn . T mt bn cho trc, hy m s lng o ca vng bin.
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 122
T mt th v hng, lin thng cho trc, liu c th nh hng li thnh th c hng lin thng mnh?
(nh l: th v hng lin thng l nh hng c khi v ch khi mi cnh ca n nm trn t nht mt chu trnh)
HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 123
Hy tm ng i gia 2 nh bt k ca mt n th v hng lin thng.
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HCMUS 2010 Bi ging L thuyt th ng Nguyn c Tin 125