Chapter 2 Basic Concepts in Graph Theory
大葉大學 資訊工程系 黃鈴玲
2011.9








2.1 Paths and Cycles
2.2 Connectivity
2.3 Homomorphisms and Isomorphisms of
Graphs
2.4 More on Isomorphisms on Simple Graphs
2.5 Formations and Minors of Graphs
2.6 Homomorphisms and Isomorphisms for
Digraphs
2.7 Digraph Connectivity
2

3

is a walk of length 7
4

5

6

7

contain
8

Theorem 2.4
Proof Hint:
Let P be a u, v-walk of shortest length. Show that P is a u, v-path.
Ex 2.1, 2.2
9

as components
10

Theorem 2.13
For a simple graph G with n vertices and k components we have
| E ( G ) |

( n
 k )( n
 k

1 )
2
Proof
Let H
1
, H
2
, …, H k be the k components of G, and |V(H i
)| = n i for each i.
| E ( G ) |


 n
1
2




 n
2
2


  

 n k
2



1
2
( n
1
2  n
2
2    n k
2  n )
By Lemma 2.14
,

| E ( G ) |
 n
1
2  n
2
2    n k
2  n
2 
( k

1 )( 2 n
 k )
1
2
[( n
2 
( k

1 )( 2 n
 k ))
 n ]

( n
 k )( n
 k

1 )
2
Ex 2.4, 2.5
11

當 f 是 G
 G’ 的 homomorphism 時， 若 u,v 兩點在 G 中相連，則這兩點對應過去的 f(u) 與 f(v) 也在 G’ 中相連
12

Let f : G
 G’ be defined by f = (f
1
, f
2
), where
13

Example 2.17
14

15

16

Example 2.19
No!
原因 1 ：左圖有一條 edge ，一個端點連接 multiedge ，
另一端點連接 loop ，但右圖沒有此種 edge
原因 2 ：左圖只有一點 degree=2 ，它的 neighbor 的 degree
分別是 3 及 4 ，但與右圖 degree=2 節點的 neighbor degree 不等
17

Ex2.13. Show that the graphs shown in Figure 2.25 are isomorphic.
18

Observation 2.22
Def 2.24
Theorem 2.25
Ex 2.22, 2.23
19

Example 2.26
Both graphs are 3-regular with 6 vertices.
G has 3cycles, but G’ has no 3-cycles.

No!
Another reason: G is planar but G’ is not. (see Chapter 7)
20

complement not isomorphic
21

Ex2.14. Which, if any, of the graphs G
1
, G
2
, G
3
, and G
4 in Figure 2.26 are isomorphic?
22

Ex2.16. Which, if any, of the graphs G
1
, G
2
, and G
3 in Figure 2.27 are isomorphic?
23

Def 2.27
24

25

Def 2.30
26

Def 2.33
Ex 2.28
Def 2.34 The contraction ( 收縮 ) of G by e (denoted by G
 e )
( 將 e 的兩端點 u , v 黏成一點 w ，原先與 u 或 v 相連的點，都與 w 相連，
新圖的總邊數只比原圖少一條 )
27

Def 2.36 Simple contraction of G by e (denoted by G / e )
( 將 G
 e 改為 simple graph ，即去除 multiedge)
28

Def 2.37
For
G’ c sequ
1.
2.
Example 2.38
Ex 2.31, 2.32
29

f: G

G’ is a homomorphism on digraphs v f(v)
 u
G
f(u)
G’
30

31

All vertices indegree=1 outdegree=1
2 vertices indegree=1 outdegree=1
They are adjacent.
2 vertices indegree=1 outdegree=1
They are not adjacent.
32

Def 2.44
33

A digraph is called weakly connected if its underlying graph is connected.
Def 2.47
These two directed paths are not necessarily edge disjoint.
34

35

Example 2.50
36


補充題 :
Show that there is no tournament on 6 vertices all of whose vertices have the same outdegree.
37