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# Some NP-complete Problems in Graph Theory PowerPoint PPT Presentation

Lecture 31. Some NP-complete Problems in Graph Theory. Prof. Sin-Min Lee. Graph Theory. An independent set is a subset S of the verticies of the graph, with no elements of S connected by an arc of the graph. Coloring.

Some NP-complete Problems in Graph Theory

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Lecture 31

## Some NP-complete Problems in Graph Theory

Prof. Sin-Min Lee

### Graph Theory

• An independent set is a subset S of the verticies of the graph, with no elements of S connected by an arc of the graph.

### Coloring

• How do you assign a color to each vertex so that adjacent vertices are colored differently?

• Chromatic number of certain types of graphs.

• k-Coloring is NP Complete.

• Edge coloring

### Planarity and Embeddings

K4 is planar

K5 is not

Euler’s formula

Kuratowski’s theorem

Planarity algorithms

BB: III – maybe two weeks?

AG: CH. 4 and 5.

### Flows and Matchings

3

6

• Meneger’s theorem (separating vertices)

• Hall’s theorem (when is there a matching?)

• Stable matchings

• Various extensions and similar problems

• Algorithms

7

t

5

2

1

1

4

s

5

3

9

girls

boys

### Random Graphs

• Form probability spaces containing graphs or sequences of graphs as points.

• Simple properties of almost all graphs.

• Phase transition: as you add edges component size jumps from log(n) to cn.

### Algebraic Graph Theory

a

a3

a2

group

elements

a

a

• Cayley diagrams

• Adjacency and Laplacian Matrices their eigenvalues and the structure of various classes of graphs

a

1

a

generators

### Algorithms

• DFS, BFS, Dijkstra’s Algorithm...

• Maximal Spanning Tree...

• Planarity testing, drawing...

• Max flow...

• Finding matchings...

• Finding paths and circuits...

• Traveling salesperson algorithms...

• Coloring algorithms...