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Lecture 16: Graph Theory III. Discrete Mathematical Structures: Theory and Applications. 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 - PowerPoint PPT Presentation
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Lecture 16: Graph Theory III
Discrete Mathematical Structures:
Theory and Applications
10
Discrete Mathematical Structures: Theory and Applications 2
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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Discrete Mathematical Structures: Theory and Applications 3
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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Discrete Mathematical Structures: Theory and Applications 4
Graph Algorithms
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Discrete Mathematical Structures: Theory and Applications 5
Graph Algorithms
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Discrete Mathematical Structures: Theory and Applications 6
Graph Algorithms
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Discrete Mathematical Structures: Theory and Applications 7
Graph Algorithms
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Discrete Mathematical Structures: Theory and Applications 8
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Discrete Mathematical Structures: Theory and Applications 9
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10
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v known dv pv
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
0 2
1
v known dv pv
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
3 3
9 5
v known dv pv
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
8
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
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v known dv pv
v1 T 0 0
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v4 T 1 v1
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v6 T 6 v7
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From 1 to 6
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Discrete Mathematical Structures: Theory and Applications 10
Graph Algorithms
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Discrete Mathematical Structures: Theory and Applications 11
Planar Graphs and Graph Coloring
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Discrete Mathematical Structures: Theory and Applications 12
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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Discrete Mathematical Structures: Theory and Applications 13
Planar Graphs and Graph Coloring
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Discrete Mathematical Structures: Theory and Applications 14
Planar Graphs and Graph Coloring
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Discrete Mathematical Structures: Theory and Applications 15
Planar Graphs and Graph Coloring
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Discrete Mathematical Structures: Theory and Applications 16
Planar Graphs and Graph Coloring
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Discrete Mathematical Structures: Theory and Applications 17
Planar Graphs and Graph Coloring
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Discrete Mathematical Structures: Theory and Applications 18
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Discrete Mathematical Structures: Theory and Applications 19
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Discrete Mathematical Structures: Theory and Applications 20
Planar Graphs and Graph Coloring
