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
Flows and Matchings
3
6
7
5
1
s
3
2
4
1
t
5
9
girls
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boys
Meneger’s theorem (separating vertices)
BB: III –
Hall’s theorem (when is there a matching?) maybe two
weeks?
Stable matchings
Various extensions
and similar problems
AG: CH. 4 and 5.
Algorithms
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
a3
• Cayley diagrams
a
a
1
a2
a
a
a
group
elements
generators
• Adjacency and Laplacian Matrices their
eigenvalues and the structure of various
classes of graphs
Algorithms
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DFS, BFS, Dijkstra’s Algorithm...
Maximal Spanning Tree...
Planarity testing, drawing...
Max flow...
Finding matchings...
Finding paths and circuits...
Traveling salesperson algorithms...
Coloring algorithms...