离散数学复习大纲-Part_II(图论基础)

1 . (Basic Concepts of Graph Theory)
1. (Basics Concepts for Undirected Graph)
(a) G =(V,E ) (undirected graph)
i.V
ii.E V
V (vertex) E (edge) (b) V (G ) G E (G ) G |V (G )|=n G n (c)i. V E
ii. V E
(d)
V ={v 1,v 2,...,v n },E ={e 1,e 2,...,e m }e k (v i ,v j )
e k v i ,v j e k v i v j v i ,v j e k v i v j n m (e)i.
ii. (loop) e k =(v i ,v i )
iii. (multiple edges) iv. (multigraph)
v. (simple graph)
vi. (null/empty graph) N n vii. (complete graph) n K n viii.
(f) (degree) G =(V,E ) v d (v ) v d (v ) 2 1 , k k - (g) G =(V,E )
v ∈V d (v )=2|E |
K n 12n (n −1)
2. (Basic Properties of Graph)
(a) G V V 1 V 2 G V 1 V 2 G V 1 V 2 G V 1 V 2 G |V 1|=m,|V 2|=n K m,n (b) G H
i. V (H )⊆V (G ),E (H )⊆E (G ) H G (subgraph) H ⊆G ii. H ⊆G,H =G , H G H ⊂G
iii. H ⊆G,V (H )=V (G ) H G (spanning sub-graph)
iv. V (H )⊆V (G ) E (H ) G V (H ) H G
(induced subgraph)
v. G N n
(c) G1=(V1,E1) G2=(V2,E2)
i. G=(V,E) V=V1∪V2,E=E1∪E2 G=G1∪G2
ii. G=(V,E) V=V1∩V2,E=E1∩E2 G=G1∩G2
iii. G=(V,E) V=V1∪V2,E=E1⊕E2 G=G1⊕G2 G2 G1
iv. G1−G2=(V1,E1−E2)
v.n G G:K n−G
vi. G v G−v
vii. G e:G−e
viii. G e ij=(v i,v j):G+e ij
(d) G1(V1,E1) G2(V2,E2) V1 V2 f G1
a b G2 f(a) f(b) G1 G2 (isomorphic)
G1∼=G2
G1∼=G2
i.|V(G1)|=|V(G2),|E(G1)=E(G2)|
ii.G1 G2
iii.G1 G2
(e) G=(V,E) e k w k (weight)
G
3. (Basics Concepts for Directed Graph)
(a) D=(V,E)
i.V
ii.E V V G E
G
(b)i. G e=(u,v) E u e e u
v e e v ii.
iii. u,v∈V (u,v)∈E (v,u)∈E D
iv. D u,v∈V (u,v)∈E (v,u)∈E0
D ( )
(c) v v d+(v) v v
d−(v) v v d(v)
d(v)=d+(v)+d−(v)
(d)
v∈V d+(v)=
v∈V
d−(v)=|E|
(e)i. : D D
ii. D
D
(f)
i. G , v
Γ(v)={u|(v,u)∈E(G)}
ii. G , v
Γ+(v )={u |(v,u )∈E (G )}
iii.v
Γ−(v )={u |(v,u )∈E (G )}
4. (Algebraic Representation of Graphs)
(a) (adjacency matrix):
i. G =(V,E ) V {v 1,v 2,...,v n } G A =(a ij )(n ×n )
a ij = 1,(v i ,v j )∈E,0,(v i ,v j )/∈E.
ii. D =(V,E ) V {v 1,v 2,...,v n } D A =(a ij )(n ×n )
a ij = 1,(v i ,v j )∈E,0,(v i ,v j )/∈E.
(b) :
a ij =
w ij ,(v i ,v j )∈E,
0,(v i ,v j )/∈E.
(c) (incidence matrix)
i. G =(V,E ) V {v 1,v 2,...,v n } E ={e 1,e 2,...,e m } M F =(mG ij )(n ×m ),
m ij = 1, e j v i ,0,> e j v i .
ii. D =(V,E ) V {v 1,v 2,...,v n } E ={e 1,e 2,...,e m } M =(m ij )(n ×m ),
m ij =⎧⎨⎩0,
e j v i ,1, e j v i ,−1, e j v i .
iii. :A. n i =1m ij =0,j =1,2,...,m m j =1 n i =1m ij =0 M (D )
0 B.M (D ) 1 1 m
C. i 1 d +(v i ) 1 d −(v i )
D. 1 -1
E. ,
F. -1
2 . (Path and Circuit)
1. (Basic Concepts for Path and Circuit)
(a)
i. G P=(e i
1,e i
2
,...,e i
q
) e i
j
=(v k,v l) v k e i j+1
v l e i
j+1 P G e i
q
e i
1
P G
ii. / P
iii. / P (b)
i. G P=(v i
1,e i
1
,v i
2
,e i
2
,...,e i
q−1
,v i
q
) v i
j
v i
j+1
e i
j
P G v i
q =v i
1
P G
ii. / P
iii. / P
iv.
(c)
i.
ii. G G G c(G) G
G g(G)
iii. K n(n≥3) n 3 K n,n(n≥2) 2n
4
(d)
i. C G 3 v i v j C
(v i,v j)∈E(G) (v i,v j) C
ii. v k∈V(G) d(v k)≥3 G
(e) G
2. (connectivity)
(a)
i. G G (connected)
ii. G G ( )
(b) (connected component)
i. G H G H G
ii. G
iii. 2
(c)
(d) u,v G u,v u,v
u,v u,v d(u,v) u,v d(u,v)=∞
(e) G=(V,E)(|V|=n,|E|=m) m≥n−1
(f)
i. G=(V,E)(|V|=n), u v(u=v)
n
ii. G=(V,E)(|V|=n) u v(u=v) n
iii. G=(V,E)(|V|=n) u u n
iv. G=(V,E)(|V|=n) u u n
v. G=(V,E) n E=∅
Γk G
Γk+r k k+r
Γk+r vi. G
(g)
3. (Euler Path and Euler Circuit)
(a) G=(V,E) G (
) G ( )
G G G
(b) G G
(c) G G
(d) G 2 G
(e)
(f) G 0 2 2
4.
(a) G ( ) G ( )
H ( )
(b)H
i. G v i v j d(v i)+d(v j)≥n1 G H
ii. (Ore ) G(n≥3) v i v j d(v i)+ d(v j)≥n G H
iii. (Dirac ) G(n≥3) ≥n
2 G H
iv. G v i,v j d(v i)+d(v j)≥n G H G+(v i,v j) H
(c)
i. v i v j G d(v i)+d(v j)≥n G =G+(v i,v j)
G (v i,v j), ,
G , C(G)
ii. G C(G)
iii. (Bondy&Chv´a tal,1976) G H C(G) H
iv. G(n≥3) C(G)=K n G H
(d)
i. G G S G S
|S| ∀S⊆
V,p(G−S)≤|S| p(G−S) G−S
ii.
iii.
3 . (Tree)
1. (Basic Concepts of Trees)
(a)
i. T
ii.
iii. 1
iv.
v. G G
vi.
(b)
i. e G G =G−e G e G
ii.
iii. e=(u,v) e G
(c)
i. G=(V,E) m n
A.G
B.G
C.G
D.G m=n−1
E.G m=n−1
F.G G
G.G
ii.
iii. T
2. (Tree with Fixed Root)
(a)
i.
A.
B. r r V0 v∈V0
(r,v) r v
C. r V0 V1
V0 V1 V0 V1
V1 V2
D. V k
ii.
iii.
A. T v (u,v) u v
u v u
B. u v u v
v u
iv. v T v v
v. v v
(b)
i. T T T
ii. T
A. T m T m m
m
B. T m T m T
m
C. T m T m
T m
(c)
h m h h−1
(d)
i. i m n=mi+1
ii.m m
A.n i=n−1
m l=(m−1)n+1
m
B.i n=mi+1 l=(m−1)i+1
C.l n=ml−1
m−1 i=l−1
m−1
iii.m h m m h
3. (Binary Tree and Huffman Tree)
(a)
i. 2 ii. 0 2 (b)
i. i 2i −1 ii. h 2h −1(h ≥1) iii. T N 0 2 N 2 N 0=N 2+1
(c)
i. 0 1 ii. α1α2...αn −1αn n
α1,α1α2,α1α2...αn −1
1,2,...,n −1 A ={β1,β2,...,βm } m βi ,βj ∈A,(i =j ) A A 0 1 A iii. iv. A ={β1,β2,...,βm } βi p i l i L = m i −1p i l i
(d)
i. T t v i w i w i v i i =1,2,...,t w (T )= t i =1w i l i T l i v i T w 1,w 2,...,w t t Huffman ii.Huffman
(1) n {w 1,w 2,...,w n } n F ={T 1,T 2,...,T n } T i w i (2) F (3) F F
(4) (2) (3) F Huffman
iii. Huffman
4. (Spanning Tree)
(a)
i. T G T G T G T G ( ) T G ∀e ∈E (G ) e ∈E (T ) e T e T G [E (G )−E (T )] T T ii. G G
iii. G n m m ≥n −1
iv. G n m T G T m −n +
1 T m −n +1
v. T G T T C G E (T )∩E (C )=∅
(b)
i. BFS(Breadth-First-Search)
(1) G G
s ( ) s s 0
(2) s 1(s)
(3) 1 v 1
2(v)
(4) (3) G
(c)
T W(T) G G
(Minimum Spanning Tree,MST) ( )
(d)Kruskal ( )
i. n G=<V,E,W> m
(1) m e1,e2,...,e m T=∅
(2) e1 T e2,e3,...,e m e j T e j T
e j
(3) n−1 T G
ii. ( )T=(V,E ) G=(V,E) T e∈E−E C e(C e⊆E +e)
w(e)≥w(a),a∈C e(a=e)
iii. Kruskal G
iv. ( )Kruskal O(m+p log m) p
v.
(e)Prim
i. G=<V,E,W> n
(1) U=∅ T=∅ V v U
(2) U v v U
T
(3) (2), U=V T G
ii. ( ) V G=(V,E) e V V−V G e T iii. Prim G
iv.。