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.

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Some NP-complete Problems in Graph Theory

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Some np complete problems in graph theory

Lecture 31

Some NP-complete Problems in Graph Theory

Prof. Sin-Min Lee


Graph theory

Graph Theory


Some np complete problems in 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

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

Planarity and Embeddings

K4 is planar

K5 is not

Euler’s formula

Kuratowski’s theorem

Planarity algorithms


Flows and matchings

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

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

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

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...


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