Lecture 16: Graph Theory III

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Lecture 16 Graph Theory III

• Discrete Mathematical Structures
• Theory and Applications

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Learning Objectives

• Learn the basic properties of graph theory
• Learn about walks, trails, paths, circuits, and
  cycles in a graph
• Explore how graphs are represented in computer
  memory
• Learn about Euler and Hamilton circuits
• Explore various graph algorithms
• Examine planar graphs and graph coloring

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Graph Algorithms

• Graphs can be used to show how different
  chemicals are related or to show airline routes.
  They can also be used to show the highway
  structure of a city, state, or country.
• The edges connecting two vertices can be assigned
  a nonnegative real number, called the weight of
  the edge.
• If the graph represents a highway structure, the
  weight can represent the distance between two
  places, or the travel time from one place to
  another.
• Such graphs are called weighted graphs.

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Graph Algorithms
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Graph Algorithms
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Graph Algorithms
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Graph Algorithms
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v1 F 0 0
v2 F 99999999 0
v3 F 99999999 0
v4 F 99999999 0
v5 F 99999999 0
v6 F 99999999 0
v7 F 99999999 0
v known dv pv
v1 T 0 0
v2 F 2 v1
v3 F 99999999 0
v4 F 1 v1
v5 F 99999999 0
v6 F 99999999 0
v7 F 99999999 0
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v1 T 0 0
v2 F 2 v1
v3 F 3 v4
v4 T 1 v1
v5 F 3 v4
v6 F 9 v4
v7 F 5 v4
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v1 T 0 0
v2 T 2 v1
v3 F 3 v4
v4 T 1 v1
v5 F 3 v4
v6 F 9 v4
v7 F 5 v4
v known dv pv
v1 T 0 0
v2 T 2 v1
v3 T 3 v4
v4 T 1 v1
v5 F 3 v4
v6 F 8 v3
v7 F 5 v4
v known dv pv
v1 T 0 0
v2 T 2 v1
v3 T 3 v4
v4 T 1 v1
v5 T 3 v4
v6 F 8 v3
v7 F 5 v4
v known dv pv
v1 T 0 0
v2 T 2 v1
v3 T 3 v4
v4 T 1 v1
v5 T 3 v4
v6 F 6 v7
v7 T 5 v4
v known dv pv
v1 T 0 0
v2 T 2 v1
v3 T 3 v4
v4 T 1 v1
v5 T 3 v4
v6 T 6 v7
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From 1 to 6
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Graph Algorithms
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Planar Graphs and Graph Coloring
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Planar Graphs and Graph Coloring

• A graph is a planar graph if and only if it has a
  pictorial representation in a plane which is a
  plane graph. This pictorial representation of a
  planar graph G as a plane graph is called a
  planar representation of G.
• Let G denote the plane graph in Figure 10.111.
  Graph G, in Figure 10.111, divides the plane into
  different regions, called the faces of G.

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Planar Graphs and Graph Coloring
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Planar Graphs and Graph Coloring
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Planar Graphs and Graph Coloring
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Planar Graphs and Graph Coloring
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Planar Graphs and Graph Coloring
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Planar Graphs and Graph Coloring