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Chapter 10 Graphs 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.

a

b

c e f

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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
**

**D
**

**C
B
**

**1****2**

**4**

**3**

7

**6****5**

**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
**

**D
**

**C
B
**

**1****2**

**4**

**3**

7

**6****5**

*Translation into a graph problem*: Land masses are “nodes”.

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The Bridges of Koenigsberg

**1****2**

**4**

**3**

7

**6****5**

*Translation into a graph problem***: Bridges are “arcs.”
**

**A
**

**C
**

**D
**

**B
**

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The Bridges of Koenigsberg

**1 2
**

**4
**

**3
**

**7
**

**6 5
**

**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.
**

**A
**

**C
**

**D
**

**B
**

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Adding two bridges creates such a walk

**A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9, B, 2, A
**

**1 2
**

**4
**

**3
**

**7
**

**6 5
**

**A
**

**C
**

**D
**

**B
**

**8
**

**9
**

**Here is the walk.
**

**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

**1 2
**

**4
**

**3
**

**7
**

**6 5
**

**A
**

**C
**

**D
**

**B
**

**8
**

**9
**

**The degree of
a node is the
number of
incident arcs
**

**6
**

**4
**

**4
**

**4
**

**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).
**

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