Graph Decomposition

1
Graph Decomposition

• Modular decomposition
• Join decomposition

2
Graph decomposition

• A basic divide-and-conquer strategy
• After the decomposition, the prime graphs might
  possess certain desirable properties
• The decomposition tree (often unique) can
  sometimes be used to compare two graphs

3
Modular decomposition
4
Modular Decomposition

• A Module is a set of vertices that are
  indistinguishable from outside.

Not a module
5
Modules of a Graph

• A module M is a subset of v(G) s.t. each vertex v
  in V \ M there are either no edges or all
  possible edges.

M
fully connected
Can reduce M into a single vertex (The reduction
step) and separate M from the remaining subgraph
6
The Types of Modules

• Let GM be the graph induced on vertex set M
• Module M is a connected iff GM is connected.
• Module M is complement connected iff the
  complement of GM is connected.
• By the definition of GM, there are three types
  of modules.
• A Parallel module
• A series module
• A neighborhood module

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A Parallel module

• An unconnected module is called a parallel module.

2
8
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M
13
10
11
12
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A series module

• A connected module whose complement is
  unconnected.

2
13
10
11
12
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A neighborhood module

• A connected module whose complement is also
  connected.

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7
4
3
6
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Modular Decomposition
5
7
4
3
6
1
G1
2
8
1
11
2
12
13
9
13
10
11
12
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Modular Decomposition
5
7
4
3
6
1
2
8
9
13
10
11
12
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Modular Decomposition
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7
4
3
6
1
2
8
9
13
10
11
12
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Modular Decomposition
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7
4
3
6
1
2
8
9
13
10
11
12
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Modular Decomposition
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7
4
3
6
1
2
8
9
13
10
11
12
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Join decomposition
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Join Decomposition
13
14
12
10
1
3
11
2
4
5
8
7
6
9
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Join Decomposition
13
14
12
10
1
3
11
2
4
5
8
7
6
9
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Join Decomposition
13
14
12
10
1
3
11
2
4
5
8
7
6
4
5
8
7
6
9
9
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Join Decomposition
13
14
12
10
1
3
11
2
3
4
5
8
7
6
9
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Join Decomposition
13
14
12
10
1
3
11
2
1
4
5
2
8
7
6
9
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Join Decomposition
13
14
12
10
1
G1
3
11
2
G4
G2
Q2
Q3
4
5
G3
8
7
6
9