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 formulaKuratowski’s theoremPlanarity algorithms
Flows and Matchings
• Meneger’s theorem (separating vertices)• Hall’s theorem (when is there a matching?)• Stable matchings• Various extensions and similar problems• Algorithms
st5
3
6
1
72
4
9
3
1 5
girls boys
BB: III – maybe two weeks?
AG: CH. 4 and 5.
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
• Cayley diagrams
• Adjacency and Laplacian Matrices their eigenvalues and the structure of various classes of graphs
1 a
a2a3
a
a
a
a
groupelements
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...
