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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. 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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  1. Lecture 31 Some NP-complete Problems in Graph Theory Prof. Sin-Min Lee

  2. Graph Theory

  3. 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.

  4. 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

  5. Planarity and Embeddings K4 is planar K5 is not Euler’s formula Kuratowski’s theorem Planarity algorithms

  6. 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

  7. 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.

  8. 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

  9. 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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