Introduction to Mathematics Jordan Davidson
Professor Swets MCS 212
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Course Topics5
1. – Paths and Networks
2. – Voting Methods
3. – Game Theory
4. – Apportionment
5. – Probability
Thursday January 18, 2007 Section 6.1, Problems 8, 12, 16, 20, and 26.
Graph Theory Graph has only dots and lines. Consists of vertices and edges. Edges may
only meet at vertices.
Terminology: Even Vertices – attached to an even number of edges is
even
.
Odd Vertices A vertex attached to an odd number of edges is
odd
.
An edge that starts and ends at the same vertex is a
loop
.
A
path
is a collection of edges, in order, that starts at some vertex and
ends at a (possibly the same) vertex.
We usually mark the edges in some way to indicate the order they are
traversed.
If the starting and ending vertices are the same, the path is a circuit.
If for
any
two vertices in a graph there exists a path starting at one of the
vertices and ending at the other, the graph is
connected
.
If there exist two points in a graph for which there is not path starting at
one of the vertices and ending at the other, the graph is
disconnected
.
A
component
of a graph is a connected part of a graph.
A
Bridge
is an essential edge that connects the graph. If taken away, the
graph becomes disconnected.
So a bridge is an edge which, if it were removed, would increase the
number of components in a graph.
A path that includes each edge in a graph exactly once is an
Euler
or
Eulerian Path
.
A circuit that includes each edge in a graph exactly once is an
Euler
or
Eulerian Circuit
.
Euler’s Theorem
 A graph has an Euler circuit if an only if
1.
It must be connected; and
2.
every vertex is even.
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View Full DocumentA graph has an Euler path if and only if
1. It is connected; and
2. it either has no odd vertices or exactly two odd vertices, and Euler path must
start at one of them and end at the other.
Fleury’s Algorithm
1.
Make sure an Euler circuit exists (graph connected and all vertices even).
2.
Start at any vertex.
3.
Travel through and mark an edge if either
a.
It is not a
bridge
for the untravelled part of the graph; or
b.
There is no other alternative
4.
Continue until done.
Thursday, January 18, 2007
 Dr. Swets –
Eulerization
 Identify all the odd vertices. (there will be an even number of them.)
Match the odd vertices in nearby pairs.
Duplicate the edges along the shortest path between each pair of vertices
you matched up.
Note: You may only duplicate existing edges never create a new edge.
Applications of Euler Circuits
The uses of them include
•
Garbage collection routes
•
Mail Delivery
•
Bus routes
•
Snow plows
•
Street sweeping
The Traveling Salesman Problem
Here is a graph with five cities (Vertices) and you want to find the cheapest way to visit
all five cities
Hamilton Circuit
 A circuit that begins at a vertex, passes through every other vertex
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 Spring '08
 Swets
 Math, Voting system, Thursday, Natural Quota

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