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Some NP-complete Problems in Graph Theory

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Lecture 31

Some NP-complete Problems in Graph Theory

Prof. Sin-Min Lee

- 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.

- 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

K4 is planar

K5 is not

Euler’s formula

Kuratowski’s theorem

Planarity algorithms

BB: III – maybe two weeks?

AG: CH. 4 and 5.

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

- 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.

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

- DFS, BFS, Dijkstra’s Algorithm...
- Maximal Spanning Tree...
- Planarity testing, drawing...
- Max flow...
- Finding matchings...
- Finding paths and circuits...
- Traveling salesperson algorithms...
- Coloring algorithms...