Graph Theory

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Graph Theory
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Simple Graphs

• ??????????????????????????????????????????????????
  ??????????????????????????????????????????????????
  ???????? ???????????????????????????????????????
  ????????????? ??????????????????? ???????????????
  ???
• ??????????????????????????????(simple graph)
  ??????????????? ??????????????????????????(symm
  etric),
• ????????????(irreflexive)
• ????????????? G(V,E)??????????
• ?????????(vertices)??????? V
• ??????????(edges)??????? E ?????????????????????
  ?????????? u,v ? V

Visual Representationof a Simple Graph
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Multigraphs

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

???? e1 ??? e2 ??????????????????(multiple)????
parallel ??? f(e1)f(e2)
Paralleledges
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Pseudographs

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

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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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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 ????

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Adjacency Matrices
c
d
e
a
b

• ????????? ??????????????(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 ?????????????

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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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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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