Graph Theory – Polygons as Networks
Network theory is a branch of topology founded by Leonhard Euler about 250 years ago. Euler worked on
two topological problems 100 years before topology had been so named. These two problems founded
network theory. One of these problems, the famous puzzle of the Konigsberg bridges, will be discussed
later.
(the following is from the textbook, Glencoe Geometry by Cindy Boyd (1998), beginning on page 559)
The international computer information system, Internet, was called the information highway in the early
1990s. It is the networking of computers, connections, and information. Internet is composed of nodes that
node
allow for information to be transferred from one computer site to another.
The diagram at the right represents a network. Such a diagram
illustrates a branch of mathematics called graph theory in networking,
the points are called nodes, and the paths connecting the nodes are
called edges. Edges can be straight or curved.
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edge
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Straight edges can be used to form closed or open graphs. If an edge of a closed graph intersects exactly
two other edges only at the endpoints, then the graph forms a polygon.
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Closed graphs
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Open graphs
However, not all pairs of nodes are connected by an edge in some networks. A network like this is called
incomplete. Therefore, a complete network has at least one path between each pair of nodes.
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Y
W
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incomplete
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L
F
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O
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N
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complete
H
You have probably seen puzzles that ask you to trace over a figure without lifting your pencil and without
tracing any lines more than once. In graph theory, if all nodes can be connected and each edge of a network
can be covered exactly once, then the network is said to be traceable.
The degree of a node is the number of edges that are connected to that node. The traceability of a network
is related to the degrees of the nodes in the network. An odd node is a node that is the endpoint of an odd
number of edges. An even node is a node that is the endpoint of an even number of edges.
It is possible to tell whether a network is traceable or not without actually trying to trace it. Euler
discovered this in his attempt to solve the problem of the Konigsberg bridges (which he proved was
impossible). The key lies in knowing how many odd or even nodes the network contains. Why? Consider a
given network that is traceable and consider one of its nodes that is neither the start nor the end of a
journey through the network. Think of this – to get to such a node you must travel to it (since you aren’t
starting there). This means there is one edge needed for the trip to the node. Now, since you are not
finishing at this node, you must leave it – this requires another edge to leave by. Is such a node always odd?
Always even? Or does it vary?
If the beginning node and the finishing node are the same node, then it is an even node. We determined in
the previous paragraph that all nodes that were not the beginning or finishing node would have to be even
nodes. In conclusion, all the nodes of a “type 1” network are even and any one of them could be the beginning
and/or finishing node(s).
If the beginning node is different from the finishing node, they would both have to be odd or both have to
be even. (If they are both even, it falls in Type 1 above.) If they are both odd and, as we’ve previously
determined, all other points are even, to draw this network, we start at one odd node and end at the other.
In summary:
1) A network is traceable on if it contains no or exactly two odd nodes.
2) If all the nodes of a network are even, a traceable trip may begin and end at any node.
3) If a network contains exactly two odd nodes, it is traceable and the journey must begin at one of the
odd nodes and end at the other.
Determine if each of the following networks is traceable and complete. If it is traceable, identify the
beginning and ending nodes.
1.
8.
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6.
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9.
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4.
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10.
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7.
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11.
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12.
13.
14.
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16.
17.
The Seven Bridges of Konigsberg
In 1735, Leonhard Euler used networks to solve a famous problem about the bridges in the city of
Koenigsberg, which is now Kaliningrad. The center of this city was on an island in the middle of the river
Pregel. The island was connected by four bridges to the banks of the river and by a fifth bridge to another
island, which was joined to the rest of the city by two more bridges. The people of Konigsberg wondered if
it were possible to travel through the city and cross each of the seven bridges only once each. After many
had tried it and had been unsuccessful, most people decided that it couldn’t be done, but they didn’t know
why.
Leonhard Euler, a great Swiss mathematician, was able to prove that it couldn’t be solve. This attempt
developed the theory of networks.
The method Euler used to solve the Konigsberg bridge
question was much like that we just used in learning about
networks. He began by utilizing what was to become a
property of topology – he reduced the land masses to
points (nodes) and stretched the bridges to edges thereby
forming a network.
A
Land A
1
2
Land B
3
5
4
Land D
1
6
2
B
Land C
A
6
1
5
7
4
3
D
7
C
B
2
4
3
6
5
C
7
D
Now using the properties of networks that we just developed, you can see that the network resulting from
the Konigsberg bridge map contains four odd nodes making it impossible to travel in a traceable fashion. The
only solution Euler could find for the people of Konigsberg was to put an eighth bridge between any two of
the four land masses. This would turn two of the four odd nodes into even nodes and it would be a network
of type 2.
Applications
Today, the study of networks and related theory has been expanded leading to many applications. Managers
have networks that represent both time and interrelationships of the many resources scheduled for a
project; and for proper planning, organization, and optimization of resources, they need to know the longest
and shortest paths through that network. In the area of communications, both the telephone company and
large corporations using leased telephone lines are concerned about the shortest paths and least-cost paths
from one point (city) to another. A manufacturer may want to know where to locate new plants and
warehouses in order to minimize the cost of shipping raw materials to plants and finished products from
plants to warehouses, wholesalers, and customers. The theory can also be applied to study processes in
chemistry, electronics, and aeronautics. Most problems solved by network theory involve so many points and
such volumes of data that their solution would not be practical without the use of computers.
An entertaining application of network theory similar to the Konigsberg bridge puzzle is these floor plans.
The challenge is to go through each door of the plan only once. Treat the room as Euler did the land masses
(nodes) in his solution to the Konigsberg bridge question and doors as he did the bridges (edges).
In each of the following floor plans, determine if it is possible to go through each of the doors only once. If
it is, determine if you must begin and end in certain rooms (identify those rooms). Note: The “outside” is
considered a room.
(a)
(c)
(b)
(d)