Graph Theory

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Graph Theory

• Chapter 10 Independence, Dominance, and
  Matchings

???? ????? ??? 2010.12
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Contents

• 10.1 Independence of Vertices
• 10.2 Domination of Vertices
• 10.3 Matchings in Graphs

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10.1 Independence of Vertices
Definition 9.18
Definition 10.1
4
Note 10.3
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Example 10.4 (Frequency Assignment
Problem)9??????(u1u9),??????????,????????????
u3
u5
u4
u2
u1
G
u6
u7
u8
u9
5?
Observation 10.6 For a graph G, the
independence number a(G) satisfies
.
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10.2 Domination of Vertices
Definition 10.11
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Example 10.13 (????police station??)??(u1u9),???
?,police station??????????,?????police station?
u5
u3
u4
u2
u1
G
u6
u7
u8
u9
3?
Observation 10.16 For a graph G, the
domination number g (G) ? a (G).
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10.3 Matchings in a Graph
Example 1.3 (Job Assignment Problem)Job J1J5,
Applicants A1A7, Job?Applicant?????????????????,
?????????
Jobs
qualified
Applicants
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Example 10.24 (Weighted Job Assignment
Problem)Job J1J5, Applicants A1A7,
Job?Applicant????????????????????(????????),?????
????????
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Definition 10.26
perfect matching
not a matching
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a
b
M ab, cd, ef is a maximum matching M
bc, de is maximal, not maximum.

c
d
e
f
Maximum (??matching???edge?) ? Maximal
(?matching??????????matching)
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Remark 10.28
Definition 10.30
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An M-augmenting path P of G
?M
?M
?M
?M
?M
?M
P?????M???M?????,M???????
Theorem 10.32
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?? Maximum Matchings in Bipartite Graphs

• Algorithm (Maximum Matching Algorithm for
  Bipartite graphs)To determine a maximum
  matching in a bipartite graph G with V(G)v1,
  v2, , vp and an initial matching M1.
• 1. i ?1, M ? M1
• 2. If i lt p, then continue otherwise, stop, M
  is a maximum matching now.
• 3. If vi is matched, then i ? i 1 and return to
  Step 2otherwise, v ? vi and Q is initialized to
  contain v only.
• 4. 4.1 For j 1, 2, , p and j ? i, let
  Tree(vj)F. (??vj??alternating tree?)
  Also, Tree(vi)T.

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• 4.2 If Q? , then i ? i 1 and return to step
  2 otherwise, delete a vertex x from Q and
  continue.
• 4.3 4.3.1 Suppose that N(x)y1, y2, ,
  yk. Let j ? 1. 4.3.2 If j ? k, then y ? yj
  otherwise, return to Step 4.2.
• 4.3.3 If Tree(y)T, then j ? j 1 and
  return to Step 4.3.2 otherwise,
  continue. 4.3.4 If y is incident with a
  matched edge yz, then Tree(y)?T,
  Tree(z)?T, Parent(y)?x, Parent(z)?y
  and add z to Q, j ? j 1, and return to Step
  4.3.2. Otherwise, y is a single
  vertex (???!) and we continue.
  4.3.5 Use array Parent to determine the
  alternating v-x path P in the tree.
  Let P ? PUxy be the augmenting
  path.
• 5. Augment M along P to obtain a new matching M.
  Let M ? M, i ? i 1 , and return to step 2.

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Example
i1, x1 is matched.
x1
y1
i2, vx2
x2
y2
x3
x5
x4
x1
x6
x3
y3
x4
y4
y5
x5
x6
y6
???Initial matching M
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Example (Fig 6.6)
x1
y1
i3, x3 is matched.
x2
y2
i4, x4 is matched.
x3
y3
x4
y4

y5
x5
i12, y6 is matched.
x6
y6
New matching M
M x2y6, x5y4, x1y1, x4y3, x3y2, x6y5 is
maximum.
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Homework

• Let G be the given bipartite graph. The edges of
  a matching M are shown in bold. Beginning with M
  and using the Maximum Matching Algorithm for
  Bipartite graphs to find a maximum matching for
  G.

v1
v2
v3
v4
v5
v6
v10
v13
v12
v11
v9
v8
v7