Graph Theory

1
Graph Theory
2
What are Graphs?
Not

• ???????????????????????????????????????????
  ????????????????????????????????????
  ???????????(coordinate)
• ?????????discrete mathematics???????????????????
  ??????????????????????????????????????????????????
  ???????????????????????????????????????
• ???????????????????????????????????????????
  ????????????? ??????????????????? ???????????????
  ???

3
Simple Graphs

• ????????????????????? R ????????????????????(symme
  tric), ?????????(irreflexive)
• ?????????????(simple graph) G(V,E)??????????
• ??? V ??????(vertices)
• ??? E ???????(edges) ?????????????????????????
  u,v ? V, ?????? uRv

Visual Representationof a Simple Graph
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Example of a Simple Graph

• ??? V ????????????????????????????????
• ????, VFL, GA, AL, MS, LA, SC, TN, NC
• ??? Eu,vu ?????????????? v
• FL,GA,FL,AL,FL,MS, FL,LA,GA,AL,AL,
  MS, MS,LA,GA,SC,GA,TN,
  SC,NC,NC,TN,MS,TN, MS,AL

NC
TN
SC
MS
AL
GA
LA
FL
5
Multigraphs

• ????????????????????? ????????????????????????????
  ???????????????????
• ????????????(multigraph) G(V, E, f )
  ??????????????????? V ,?????????? E
  ????????????fE?u,vu,v?V ? u?v
• ???????? ???? ??????????? ????????????????????????
  ?????????

Paralleledges
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Pseudographs

• ???????????????????? ?????????????????????????????
  ??????? (R ????????????????????)
• ?????????(pseudograph) G(V, E, f )
  ??????fE?u,vu,v?V ???? e?E ??????? ???
  f(e)u,uu
• ???????? ???? ?????? ?????????????
• ???????????????????????????????
• ???????????????????????????

7
Directed Graphs

• ???????????????????????? R ???????????????????????
  ????
• ???????????????(directed graph) (V,E)
  ??????????????????? V ????????????????????? E
  ????? V
• ???????? ???? V ?????????????,E(x,y) x
  ??? y

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Directed Multigraphs

• ???????????????????? ?????????????????????????????
  ???????????????????????????????????
• ???????????????????????(directed multigraph)
  G(V, E, f ) ??????????????????? V ?????????? E
  ???????????? fE?V?V
• ???????? ????., V???????,E???????????? WWW
  ?????????? ???????????????????????

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Types of Graphs Summary
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Graph Terminology

• ??? G ????????????????????????????????????????? E
  ??? e?E ?????? u,v ??????? ??????????????
• u, v ?????????(adjacent) ?????????????????(neighbo
  rs) ????????????(connected)
• ???? e ?????????????????(incident)
  ????????????????? u ?????? v
• ???? e ??????(connects) u ??? v ??????
• ??? u ?????? v ???????????(endpoints) ??????? e

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Example
Edge incidentwith b,d
e
g
a
b
d
f
AdjacentVertices
c
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Degree of a Vertex

• ??? G ?????????????????????, ??? v?V
• ?????(degree) ??? v ???????????? deg(v),
  ?????????????????????(incident)??????????
  (?????????????????????????????????)
• ????????????????? 0 ???????? ??????(isolated)
• ????????????????? 1 ???????? pendant

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Graph Terminology

• Example ?????????????????????????????? ?????????
  pendant ????????????????????????
  ?????????????????????????????????

Solution ??? f ????????????? ?????? a, d ??? j
???? pendant ???????????????????????????? g ????
deg(g) 5 ???????????????????
?????????(pseudograph) (??????????? ???????????)
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Graph Terminology

• ???????????????????????????? ?????????????????????
  ???????????????????????????????????????????????

Result ??????????? 9 ???? ????????????????????
18 ?????????????????? ????????????????????????????
??????? ??????????????????????????????????????????
????????
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Handshaking Theorem

• ??? G ????????????????????? ????????????? V
  ????????????? E ???????
• Corollary ?????????????????????
  ???????????????????????????????????????
• ???????? ???? ???????????????? 10 ???
  ??????????????? 6 ???????????????????????????
• ??? ????????????????????????????????? 6?10 60
  ??? Handshaking Theorem ???????? 2e 60 ???????
  ?????????????????? 30 ????

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Directed Degree

• ??? G ???????????????????, v ????????????? G
• ?????????(in-degree) ??? v, deg?(v),
  ??????????????????????????? v
• ????????(out-degree) ??? v, deg?(v),
  ?????????????????????????? v
• ?????(degree) ??? v, deg(v)?deg?(v)deg?(v),
  ?????????????????????????????????? v
• Directed Handshaking Theorem ??? G
  ????????????????????????????????????? V
  ????????????????? E ???????

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Directed Degree

• Example ?????????????????????????????? a, b, c,
  d ??????????????

deg-(b) 4 deg(b) 2
deg-(a) 1 deg(a) 2
deg-(d) 2 deg(d) 1
deg-(c) 0 deg(c) 2
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Special Graph Structures

• ????????????????????????????????
• Complete graphs Kn
• Cycles Cn
• Wheels Wn
• n-Cubes Qn
• Bipartite graphs
• Complete bipartite graphs Km,n

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Complete Graphs

• ????????????????? n?N, ???????????(complete
  graph) ????? n ???, Kn, ????????????????????? n
  ??? ?????????????????????????????????? ?u,v?V
  u?v?u,v?E

K1
K4
K3
K2
K5
K6
????????? Kn ?? ????
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Cycles

• ?????????????? n?3, ????(cycle)????? n ???, Cn,
  ???????????????????? Vv1,v2, ,vn ???
  Ev1,v2,v2,v3,,vn?1,vn,vn,v1

C3
C4
C5
C6
C8
C7
??????????????????????????? Cn?
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Wheels

• ?????????????? n?3, ?????(wheel) Wn,
  ????????????????????????????????? Cn
  ?????????????? vhub ?????????? n ???? vhub,v1,
  vhub,v2,,vhub,vn

W3
W4
W5
W6
W8
W7
??????????????????????????? Wn?
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n-cubes (hypercubes)

• ????????????????? n?N, ????????(hypercube) Qn
  ????????????????????????????????????????? Qn-1
  ?????????????????????????????? ??? Q0 ?? 1 ???

Q0
Q1
Q4
Q2
Q3
?????????????????? 2n ??????????????????????????
????
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Bipartite Graphs

• ????? ???? G(V,E) ???????????????(bipartite)
  ?????????? V V1 ? V2 ?????? V1nV2? ??? ?e?E
  ?v1?V1,v2?V2 ev1,v2
• ?????????????????????????????????
• ????????????????????????????????????????????????
• ????????????????????????????????????????????????

V2
V1
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Bipartite Graphs

• Example I ???? C3 ???????????????(bipartite)?????
  ???

No, ?????????????????????????????
??????????????????????????????????????????????????
?????
Example II ???? C6 ???????????????(bipartite)????
????
Yes, ???????????????????? C6 ?????????????????????
????????
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Complete Bipartite Graphs

• ?????? m,n?N, ??????????????????(complete
  bipartite graph) Km,n ?????????????????? V1
  m, V2 n, ??? E v1,v2v1?V1 ? v2?V2
• ????????? m ???????????????? ???
• n ??????????????? ???
• ???????????????????????????????
• ???????????????????????

K4,3
Km,n ?? _____ ?????? _____ ????
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Subgraphs

• ????????(subgraph) ??????? G(V,E) ???????
  H(W,F) ?????? W?V ??? F?E

K5
??????????? K5
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Graph Unions

• ??????(union) G1?G2 ???????????????? G1(V1, E1)
  ??? G2(V2,E2) ???????????????? (V1?V2, E1?E2)

?
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Graph Representations Isomorphism

• ??????????(Graph representations)
• Adjacency lists
• Adjacency matrices
• Incidence matrices
• ?????????????????(Graph isomorphism)
• ?????????????????????(isomorphic) ??????????
  ?????????????????????????????????????
  ???????????????????????????????

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Adjacency Lists

• ?????????? 1 ?????????????? ??????????????????????
  ?????????????????

b
a
d
c
e
f
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Adjacency Lists
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Adjacency Matrices

• ???????? Aaij, ?????? aij ???? 1 ??? vi, vj
  ??????????????? G, ??????? 0 ?????????????????????
  ????????
• ?????????????????? ???????????????????????????????
  1 ???????????????????????????????????????? 1 ????

• ????????? ??????????????(Adjacency matrices)
  ????????????????????? ????????????????????????

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Incidence Matrices

• ????? ??? G (V, E) ??????????????????????
  ?????? V n ????????????????????????? G ??????
  v1, v2, , vn ??? e1, e2, , em
• ??????????????(incidence matrix) ??? G
  ???????????????????????????????
  ?????????????????????? 0-1 ???? n?m ???????? 1
  ?????????? (i, j) ????????? ej ????????? vi, ???
  0 ??????????????????????????
• ??????????????? ?????????????? M mij,
• mij 1 ????????? ej ???????????? vi mij
  0 ???????????? ej ???????????? vi

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Incidence Matrices
e1 e2 e3 e4 e5 e6
a b c d e
e1
e6
e3
e5
e2
e4
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Incidence Matrices

• Example ????????????????????? M ??????? G
  ???????????????? a, b, c, d ??????? 1, 2, 3, 4,
  5, 6?

Solution

• ????????? ???????????????????????????????????
  ?????????????????? 1?????? ???????????????????????
  ??????????????????????? ??????????????????????????
  ????????? 1 ?????????????

37
Graph Isomorphism

• ?????
• ????????????? G1(V1, E1) ??? G2(V2, E2)
  ??????????(isomorphic) ?????????? ? bijection f
  V1?V2 ?????? ?a,b?V1, a ??? b ???????????????
  G1 ?????????? f(a) ??? f(b) ??????????????? G2
• f ???????????????????????????????????????????????
  ?? ????????????????????????????????

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Graph Invariants under Isomorphism

• ????????????????? G1(V1, E1) ?????????????????
  G2(V2, E2) ??????????????????????????????????????
  ??????(invariants), ??????????????????????????????
  ?????????????????? ??????????????????????
• ?????? V1V2, ??? E1E2
• ?????????????????? n ????????????????????????
• ???????????? g ???????????? ??????????????????????
  ??????????????????????????????? g
• ????????????????????????????????????????????????
  ???????????????????? ??????????????????????????
  ????????????????????????????????????????????????
  ???????????????????????

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Graph Isomorphism-Example

• ???????? ????????????
• ??????????
• ??????? ??????????????????????????

2
2
3
3
1
1
5
4
5
4
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Graph Isomorphism-Example

• ????????? ????? f (1) 1 ????????????????????????
  ???????????????????????

2
2
1
1
3
3
5
4
5
4
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Graph Isomorphism -Example

• ????????? ????? f (1) 1 ????????????????????????
  ??????????????????????? ??????????? 3, ?????????
  2 ??????????????? f (2) 3

2
2
3
1
1
3
5
4
5
4
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Graph Isomorphism -Example

• ????????? ????? f (1) 1 ????????????????????????
  ??????????????????????? ??????????? 3, ?????????
  2 ??????????????? f (2) 3 ??????????? 5
  ??????????????? f (3) 5

2
2
3
3
1
1
5
4
5
4
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Graph Isomorphism-Example

• ????????? ????? f (1) 1 ????????????????????????
  ??????????????????????? ??????????? 3, ?????????
  2 ??????????????? f (2) 3 ??????????? 5
  ??????????????? f (3) 5 ??????????? 2
  ??????????????? f (4) 2

2
2
3
3
1
1
5
4
4
5
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Graph Isomorphism-Example

• ????????? ????? f (1) 1 ????????????????????????
  ??????????????????????? ??????????? 3, ?????????
  2 ??????????????? f (2) 3 ??????????? 5
  ??????????????? f (3) 5 ??????????? 2
  ??????????????? f (4) 2 ??????????? 4
  ??????????????? f (5) 4

2
2
3
3
1
1
5
4
5
4
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Graph Isomorphism-Example

• ????????? ????? f (1) 1 ????????????????????????
  ??????????????????????? ??????????? 3, ?????????
  2 ??????????????? f (2) 3 ??????????? 5
  ??????????????? f (3) 5 ??????????? 2
  ??????????????? f (4) 2 ??????????? 4
  ??????????????? f (5) 4 ???????????????????????
  f (1) 1 ???????????????????????????????????
  ????????????????????????????????? f
  ????????????????????????????????????????????

2
2
1
1
3
3
5
4
5
4
?????????????????????
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Isomorphism Example

• ?????????????????????????????????????

?????? Yes, ?????????????????????
??????????????????????????????????????????????????
????????????????????? ??????? b ????????????????
a, c ?????????????????? ????????????????????????
??????? f ????????????????????????????????????????
f(a) e, f(b) a, f(c) b, f(d) c, f(e) d
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Isomorphism Example

• ???????????????????????????? ?????????????????????
  ?????????????????????? ???????????????????????????
  ?????????????

d
b
b
a
a
d
c
e
f
e
c
f
?????????????????????
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Isomorphism Example

• ?????????????????????????????????????

Solution No, ????????????????????????
?????????????????????????????????
?????????????????????????? d ????????????????
???????????????????????????????????????????????
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Are These Isomorphic?

• ???????????????????????????? ?????????????????????
  ?????????????????????? ???????????????????????????
  ?????????????

• ????????????????????????????

a
b

• ?????????????????????????????

• ????????????????????????? 2 ??????????
  (?????????????? 1 ??? ????????????? 3 ???)

d
e
c
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Connectivity

• ????(path)???????????? n ?????? u ???????? v
  ????????????????????????????????? u ???????? v
• ????????????????????????????????(circuit) ??? uv
• ???????????????????????????????
  ?????????????(connected) ??????????
  ??????????????????????????????????????????????????
  ?

• Yes

• No

• No

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Euler Hamilton Paths

• ????????????(Euler circuit) ?????? G
  ???????????????????????????????????????????? G
• ???????????????(Euler path) ?????? G
  ??????????(????)??????????????????????????????????
  ??? G
• ????????????(Hamilton circuit) ???????????????????
  ????????????????? G ??????????????????????????
• ???????????????(Hamilton path) ??????????(????)???
  ?????????????????????????? G ?????????????????????
  ?????

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Euler circuit Euler path

• ??????? ????????????(Euler path) ??????
  ??????????????????????????????????????????????????
  
• ??????? ???????????? (Euler circuit) ?????? ???
  ??????????????????????????????????????????
• ??????????????????????????????????? ????????????
  (Eulerian graph)
• ???????? ???? G1 ??????????????( Euler path) a,
  c, d, e, b, d, a, b

a
b
c
d
e
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Example

• abcdefgehia ???????? ??????????????????????
  ??????????????????????????????? bd, hd, hc ??? ci
• ??????????????? G ???????????????? ?????????????
  G ?????????????????
• abicbdchdefgehia

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Bridges of Königsberg Problem

• ????????????????????????????????(A,B,C,D)
  ???????????????????????????????????????????
  ?????????????????????????????????????????

A
D
B
C
????????????
???????????????????????????
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Euler Path Theorems

• Theorem ????????????????????????(connected
  multigraph) ???????????????? ??????????
  ??????????????????????
• Theorem ????????????????????????
  ??????????????????? (????????????????????)
  ????????????????????????? 2 ??????????????????????
  ??????
• ??????????????????????, ??????????????????????????
  ?
• ?????????????????????????????(Euler Circuit
  Algorithm)
• ?????????????????
• ???????????????????????????????
  ???????????????????????????????????????????
• ??????????????????????????
• ??????????????????????????????????????????????

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Hamilton circuitHamilton path

• ???????????? (Hamilton path) ???????
  ???????????????????????????????
  ??????????????????????????????????????
• ???????????? (Hamilton circuit) ??????? ???
  ?????????????????????????????????
  ??????????????????????????????????????
• ???????????? (Hamiltonian graph) ???
  ????????????????????????

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Hamiltonian Graph
????????????????????????????????????????????
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Hamilton path

• ??? G ?????????????????????????????????
• ???? G ????????????????? abcde ???????????????????
  ????????????? G ????????????????? ??????????? de
  2 ????? ?????????????????????????????? G
  ????????????????? ??????? G ???????????????????

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Hamilton circuit

• ??? G ????????????????????????????????? (?)
  ????????????????????????????????????????????????
  (?) ?????????????????????? ???????? G
  ????????????????

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Round-the-World Puzzle

• ?????????????(traverse) ??????????????????????????
  ? 12 ??????????????????????????????????????????

?????????? 12 ????
?????????????????? 12 ????
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Hamilton Paths

• ???????????????????? ?????????????????????????????
  ??????? ??????????????????????????? ????
  ?????????????????????????? ???????????????????????
  ???????????????? ?????????????????????????????????
  ?(????)????????

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Hamiltonian Path Theorems

• Diracs theorem ??????? G ??????????????????????
  ???????(connected, simple)????????????? n?3 ???,
  ??????????v deg(v)?n/2, ???????? G ??????????????
• Ores corollary ??????? G ???????????????????????
  ?????? ????????????? n3 ??? ??? deg(u)deg(v)n
  ????????????? ???u,v ???????????????????????? ,
  ???? ???? G ??????????????

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????????

• ????????????? G ?????????????????????????
  ????????????????

?????? ????????????? G ??????????????????????? 5
?????????????????????????????? ??????????? 3 ????
3 5/2 ??????? G ?????????????????? ????????????
????? G ????????????????
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Planar Graph

• ??????? ??????????? G ??? ????????????? (planar
  graph) ????????????????????????????????? G
  ?????????????????????????????????????????????
• ??????????????????????????????????????????????????
  ????????????????

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Example

• ????????? 3 ???? ??? ???????????3 ????
  ???????????????????????????????????????????????
  ???????? ?????? 3 ???? ???????????????????????????
  ????????????? 3 ???? ?????? ??? ????? ???????????
  ???????????????????????????????? ????????
  ???????????????? ?????????????????????????????????
  ????????????? ????????????????????????????????????
  ????????????

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Degree of a face

• ??????? ??? G ?????????????????
  ????????????????????????????? G ?????????
  ?????????????????????????????????????? ????
  (faces)
• ???????????? (degree of a face) ???????????? d(f)
  ??????? ????????????????????????? ????
  ??????????????? ????????????????????????
  ??????????????????????????????????????????????????
  ?????????? (the infinite face)

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Example

• ?????????????????? 3 ???? ??? f1 , f2 ??? f3
  ????? f3 ????????????????? d(f1) 3, d(f2)
  4, d(f3) 5 ???????? ?????????????????????????
  ????????? 12 ?????????????? 6 ????
• ?????????????? ???????????????????????????????????
  ??????????????????????????? ??????????????????????
  ??????????????????????????????????
  ???????????????????????????????
  ????????????????????????????? ????????????????????
  ?????? G ?????????????? n ???? ??????? f1, f2, ,
  fn ?????????????? e ???? ????????

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Example

• ????????????????? 10 ???? ??? f1, f2, f3,,f10
  ?????????????????? 22 ???? ???
• ??????? ??? G ????????????????????????????????????
  ? ?????????? n ??? ????????? e ???? ????????????
  f ???????? n e f 2
• ?????????????????? ??????? 14 22 10 2

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Weighted Graphs

• ??????????????? ??? ?????????????????? e ???????
  G ?????????????????????????????????
• w(e) ????????? ???????? w(e) ??? ??????? (weight)
  ??????? e ??? ??????????
• ??????? G ???? w(G) ?????? w(G) ???
  ???????????????????????????????? G

860
2534
191
1855
722
908
957
760
606
834
349
2451
1090
595
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Shortest Path Problems

• ???????????????????????????????????????????
  ?????? ?????????????????????????????????? (The
  traveling salemans problem) ?????????????????????
  ??????????????????????????????????????????????????
  ?? ?????????????????????????????????????
  ????????????????????????????????????????????????
• ??????? ?????????????????????????????????????????
  ?????????????? ???????????????????????????????? 2
  ?????????????????????????????????????????
  ??????????????????????????????????????????????????
  ??????????? ?????????????????????????
  ????????????????????????????????? ???
  ?????????????????????????????????????????????????
  ???????????????????????????????????????
  ??????????????????????????????????????

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Shortest Path Problems

• ??????????????????????????????????????????????????
  ????????? 2 ?????????????? ???????????????????????
  ???????????????????????????????????
  ??????????????????????????????????????
  ??????????????????????????????????
  ???????????????????????????????????????? ???
  ?????????? ???? ??????? (Dijkstra, Edsger Wybe)
• Dijkstras algorithm ?????????????????????????????
  ?????????????????????? a ?????? z
  ?????????????????

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Dijkstras algorithm

• procedure ??????? (G ????????????????????????????
  ???????????????? ?????????????????????????????)
• G ????? a v0, v1, , vn z ?????????? w(vi,
  vj) 8 ??? vivj ????????????? G
• begin
• for i 1 to n do
• L(vi) 8
• L(a) 0
• S Ø
• ??????????????????? ????????? a ???? 0
  ????????????????? ???? 8 ??? SØ

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Dijkstras algorithm

• while z ? S
• begin
• u ??????????????? S ??? L(u) ????????????
• S S ? u
• for v ? S
• if L(u)w(u,v)ltL(v) then L(v)L(u)w(u,v)
• ?????????? S ???????????????????? ???
  ????????????????????????????????? S
• end L(z) ?????????????????????????? a ??????
  z
• end

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Example

• ??????????????????????????? a ???????? z
  ??????????????????

?????????????? a ???? 0 ??? ?????????????????? 8
??????? L0(a) 0 ?????? L0(b), L0(c), L0(d),
L0(e) ??? L0(z)???? 8 ??? S0 Ø
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Example

• L1(b) min L0(b), L0(a) w(a, b) min 8, 0
  4 4
• L1(c) min L0(c), L0(a) w(a, c) min 8, 0
  2 2
• ?????? L1(d), L1(e) ??? L1(z) ???? 8
• ??????? S2 a, c

???????? a ????????????????????????? ?????????????
S1 ??????? S1 a ??????????????? ??????????
?????????????? a ???????
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Example

• L2(b) min L1(b), L1(c) w(c, b) min 4, 2
  1 3
• L2(d) min L0(d), L1(c) w(c, d) min 8, 2
  8 10
• L2(e) min L0(e), L1(c) w(c, e) min 8, 2
  10 12
• ?????? L2(z) ???? 8
• ??????? S3 a, c, b

• ???????????????????????????????????? c

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Example

• L3(d) min L2(d), L2(b) w(b, d) min 10, 3
  5 8
• ?????? L3(z) ???? 8
• ??????? S4 a, c, b, d

• ???????????????????????????????????? b

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Example

• L4(e) min L2(e), L3(d) w(d, e) min 12, 8
  2 10
• L4(z) min L0(z), L3(d) w(d, z) min 8, 8
  6 14
• ??????? S4 a, c, b, d, e

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Example

• L5(z) min L4(z), L4(e) w(e, z) min 14,
  10 3 13
• ??????? S5 a, c, b, d, e, z
• ?????????????????????????????? a ???????? z ???
  acbdez
• ??????????????????? 13 ?????

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Graph Coloring

• ????????????(graph coloring) ????????????????????
  ??????? ??????????????????????????????????????????
  ????????????????? ?????????????
• Chromatic number ?????????????????????????????????
  ?????????????????????
• ???? C5 ??????? Chromatic number ???? 3
• ???? C4 ,C6 ??????? Chromatic number ???? 2
• ???? ???? Cycle Cn ??????? Chromatic number ????
  3 ????? n ???????????? ?????????? Chromatic
  number ???? 2 ????? n ????????????

C6
C5
C4
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Example

• ????????????????????? Kn ???????????????????? n
  ?? ???????????????????????? K m, n ?????
  Chromatic number??? 2

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The 4-color theorem

• Chromatic number ?????????(planar graph) 4
• The Four color theorem chromatic number
  ????????????????????????????????? 4
• Example ???? G1 ?? chromatic number 3, ???? G2
  ?? chromatic number 4

G1
G2
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Application of Graph Coloring

• ????????????????????????????
• ??????????????????????????????????????????????????
  ??????????????????????
• ???????????? ?????????????????????????????????????
  ???????????????????????? 2 ???????????????????????
  ???
• ???????????????????????????????????????????????

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Example

• ???????????????????????????????????????????????
  ???????????????????????????????????? 7 ????
  (????????????????? 1, 2,,7) ?????????????????????
  ???????????????????????? ?????????????????????????
  ??????????????????????????????????
• 1-2, 1-3, 1-4, 1-7
• 2-3,2-4,2-5,2-7
• 3-4,3-6,3-7
• 4-5,4-6
• 5-6,5-7
• 6-7

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