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

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Presentation transcript:

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 K 4 is planar K 5 is not Euler’s formula Kuratowski’s theorem Planarity 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 s t girlsboys 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 1a a2a2 a3a3 a a a a group elements 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...