programming-for-dummies

Understanding Graphs 379
Figure 5-4:
The Seven
Bridges of
Königsberg
represented
as a graph.
The seven bridges of Konigsberg
The seven bridges of Konigsberg
represented as a graph
A mathematician named Leonhard Euler used a graph to prove that this was
impossible. In the seven bridges problem, one node has five bridges leading
to it (5 degrees) while the other three nodes only have three bridges leading
to it (3 degrees). Euler proved that the only way you could cross every
bridge exactly once was if a graph had, at the most, two nodes with an odd
number of bridges (degrees), and each odd-numbered degree node had to be
the starting and ending point.
Although solving a problem of walking across a bridge exactly once might
seem trivial, knowing how to solve this problem can help design truck or airplane
routes as efficiently as possible.
Finding the shortest path through a graph
After using Euler’s proof to design a graph that can be traversed in a single
path, graph theory can now help you find the shortest path through that
graph.
Book III
Chapter 5
Graphs and Trees
This problem, dubbed the Chinese Postman problem after the Chinese mathematician
Mei-Ko Kuan, involves finding the shortest route a postman can
travel to visit every node in a graph.
The answer to this problem can help airlines choose the shortest routes for
each airplane route to visit multiple cities. The shorter the route, the less
fuel needed and the lower the operating costs. Another use for the Chinese
Postman problem is finding the most efficient way to route e-mail from one
computer to another. In both cases, one node in the graph is the starting
point and a different node is the ending point.