Chapter 11 Graphs and Trees

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Chapter 11Graphs and Trees

• This handout
• Terminology of Graphs
• Eulerian Cycles

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Terminology of Graphs

• A graph (or network) consists of
• a set of points
• a set of lines connecting certain pairs of the
  points.
• The points are called nodes (or vertices).
• The lines are called arcs (or edges or links).
• Example

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Graphs in our daily lives

• Transportation
• Telephone
• Computer
• Electrical (power)
• Pipelines
• Molecular structures in biochemistry

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Terminology of Graphs

• Each edge is associated with a set of two nodes,
  called its endpoints.
• Ex a and b are the two endpoints of edge e
• An edge is said to connect its endpoints.
• Ex Edge e connects nodes a and b.
• Two nodes that are connected by an edge are
  called adjacent.
• Ex Nodes a and b are adjacent.

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Terminology of Graphs Paths

• A path between two nodes is a sequence of
  distinct nodes and edges connecting these nodes.
• Example
• Walks are paths that can repeat nodes and arcs.

a
b
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A little history the Bridges of Koenigsberg

• Graph Theory began in 1736
• Leonhard Eüler
• Visited Koenigsberg
• People wondered whether it is possible to take a
  walk, end up where you started from, and cross
  each bridge in Koenigsberg exactly once

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The Bridges of Koenigsberg
A
1
2
3
B
4
C
5
6
7
D
Is it possible to start in A, cross over each
bridge exactly once, and end up back in A?
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The Bridges of Koenigsberg
A
1
2
3
B
4
C
5
6
7
D
Translation into a graph problem Land masses
are nodes.
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The Bridges of Koenigsberg
1
2
3
4
6
5
7
Translation into a graph problem Bridges are
arcs.
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The Bridges of Koenigsberg
1
2
3
4
6
5
7
Is there a walk starting at A and ending at A
and passing through each arc exactly once?
Such a walk is called an eulerian cycle.
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Adding two bridges creates such a walk
3
8
1
2
4
6
5
9
7
Here is the walk.
A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9,
B, 2, A
Note the number of arcs incident to B is twice
the number of times that B appears on the walk.
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Existence of Eulerian Cycle
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The degree of a node is the number of incident
arcs
6
4
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Theorem. An undirected graph has an eulerian
cycle if and only if (1) every node degree
is even and (2) the graph is connected (that
is, there is a path from each node to
each other node).