Transcript Chapter 2 Basic Concepts in Graph Theory

Chapter 2 Basic Concepts in Graph Theory
大葉大學 資訊工程系 黃鈴玲
2011.9
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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
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is a walk of length 7
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contain
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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
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as
components
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Theorem 2.13
For a simple graph G with n vertices and k components we have
(n  k )( n  k  1)
| E (G ) |
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Proof
Let H1, H2, …, Hk be the k components of G, and |V(Hi)| = ni for each i.
 n1   n2 
 nk  1 2
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| E (G) |            (n1  n2    nk  n)
2  2
2 2
n  n2   nk  n2  (k 1)(2n  k )
1 2
(n  k )( n  k  1)
| E (G ) | [( n  (k  1)( 2n  k ))  n] 
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By Lemma 2.14,
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Ex 2.4, 2.5
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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
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Let f : G G’ be defined by f = (f1, f2),
where
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Example 2.17
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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
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Ex2.13. Show that the graphs shown in Figure 2.25 are isomorphic.
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Observation 2.22
Def 2.24
Theorem 2.25
Ex 2.22, 2.23
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Example 2.26
Both graphs are 3-regular with 6 vertices.
G has 3-cycles, but G’ has no 3-cycles.
 No!
Another reason: G is planar but G’ is not. (see Chapter 7)
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complement
not isomorphic
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Ex2.14. Which, if any, of the graphs G1, G2, G3, and G4 in Figure 2.26 are isomorphic?
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Ex2.16. Which, if any, of the graphs G1, G2, and G3 in Figure 2.27 are isomorphic?
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Def 2.27
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Def 2.30
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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)
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Def 2.36 Simple contraction of G by e (denoted by G/e)
(將Ge改為simple graph，即去除multiedge)
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Def 2.37
For
G’ c
sequ
1.
2.
Example 2.38
Ex 2.31, 2.32
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f: GG’ is a homomorphism on digraphs
v
f(v)

u
f(u)
G
G’
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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.
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Def 2.44
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A digraph is called weakly connected if its underlying graph is connected.
Def 2.47
These two directed paths are not necessarily edge disjoint.
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Example 2.50
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補充題:
 Show that there is no tournament on 6 vertices
all of whose vertices have the same outdegree.
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