Title: Chapter 11 Graphs and Trees
1Chapter 11Graphs and Trees
- This handout
- Terminology of Graphs
- Eulerian Cycles
2Terminology 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
3Graphs in our daily lives
- Transportation
- Telephone
- Computer
- Electrical (power)
- Pipelines
- Molecular structures in biochemistry
4Terminology 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.
5Terminology 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
6A 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
7The 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?
8The Bridges of Koenigsberg
A
1
2
3
B
4
C
5
6
7
D
Translation into a graph problem Land masses
are nodes.
9The Bridges of Koenigsberg
1
2
3
4
6
5
7
Translation into a graph problem Bridges are
arcs.
10The 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.
11Adding 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.
12Existence of Eulerian Cycle
4
The degree of a node is the number of incident
arcs
6
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).