Graphs & Networks
Route Inspection
The Chinese Postman Problem
Route Planning
Map Colouring
Where graph theory came from…
The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both
sides of the Pregel River, and included two large islands which were connected
to each other and the mainland by seven bridges.
The problem was to find a walk through the city that would cross each bridge
once and only once. The islands could not be reached by any route other than
the bridges, and every bridge must have been crossed completely every time
(one could not walk half way onto the bridge and then turn around and later
cross the other half from the other side).
Where graph theory came from…
• The Chinese Postman algorithm was
invented in 1962 and takes its name
from the Chinese mathematician Kuan
Mei-Ko who considered the problem of a
postman who has to deliver mail to
houses along each of the streets in a
particular housing estate and wants to
minimise the distance he has to walk.
• He must return to his starting position
Traversability
• A network is said to be Traversable if you can draw it
without removing your pen from the paper and
without retracing the same arc twice.
Are these Graphs Traversable?
What is the difference between
the 3 graphs?
No
Yes
Yes
To be traversable it must have 2 or 0 odd nodes
• In order to start and end at the same node the network must be
Eularian (all nodes must be even).
• If exactly two nodes have odd order, the network is still
traversable, but you start and end at different nodes (and the
network is Semi-Eularian)
To be route for a
postman it must be
closed and so have
0 odd nodes
Traversability
• A network is said to be Traversable if you can draw it
without removing your pen from the paper and
without retracing the same arc twice. It is Closed if
you can start anywhere and finish at the start point.
Are these Graphs Traversable? Are they closed?
Traversable?
Closed?
Yes
Yes
Yes
No
Yes
Yes
For each network, count the number of odd vertices and the
number of even vertices, then complete the table.
Network
Number of even
vertices
Number of odd
vertices
Traversable?
Closed?
Notes…
• If the graph in question is closed then such a
trail is easy to find. However, many graphs
are not closed and this is where the problem
becomes more interesting.
• Such a network can be made closed by linking
together pairs of odd nodes with additional
arcs. The effect of adding these extra arcs is
to make all nodes even and thus the network
is now closed.
This graph has four vertices of odd
order and so it is not closed or
traversable.
We need to pair the vertices together by adding an
extra edge to make the order of each vertex four. We
can join AB and CD, or AC and BD, or AD and BC.
In each case the graph is now closed.
[You only need to add one arc to be traversable]
If the length of each road is 1
mile, find the length of an
optimal Chinese postman
closed route for these networks
Starting and finishing at A write
down one solution such as
ABGC….
Draw extra line from B to C
9 miles: One route is AFEDCGBCBA
Draw two extra lines B to C and E to F
12 miles:
One route is: AFGEFEDCGBCBA
12 miles:
One route is: ABCEDCGBFGEFA
A groundsman at a local sports centre has to mark out the lines of
several five-a-side pitches using white paint. He is unsure as to the size
of the goal area and he decides to paint the outline as given below,
where all the distances are in metres.
He starts and finishes at the point A. Find the minimum total
distance that he must walk and give one of the
corresponding possible routes.
Draw an extra route from B to E
180 metres: One possible route: ABEBCDEFA
A local council is responsible for gritting roads.
The following diagram shows the lengths of roads, in miles, that have to
be gritted.
He starts and finishes at the point A. Find the minimum total
distance that he must drive and give one of the
corresponding possible routes.
Draw an extra route from D to E and D to C
80.5 miles: One possible route: ABFEBCDEDCA
Find the length of an optimal Chinese postman route for the
networks below. Start and end at C.
43
93
Directed Chinese Postman Problem
There are many variations to the Chinese Postman
Problem, most notably whether the roads are one-way
Taking each arc to be of equal weight, one
optimal closed Chinese Postman Tour of
this graph is:
0, 1, 3, 0, 1, 2, 3, 0, 2, 3, 0,
traversing 10 arcs.
A least cost open tour is:1
, 3, 0, 1, 2, 3, 0, 2,
which traverses only 7 arcs.
Route Planning Problem
Is it possible to plan a walk
in which each door is passed
through exactly once?
Each room is represented by a vertex point. Notice that a
vertex (point P) is needed to represent the region outside
the floor plan.
Now we need to
count the order of
each of the nodes.
Is it possible to plan a walk in which each door is
passed through exactly once?
Can you start inside or outside the building?
Graph Coloring Problem
Your mission color the entire map of South America.
1. No country may touch another country of the same color.
2. You will be charged each time you use a color
3. You must color the map as cheaply as possible.
We start with the graph representation: vertices represent
countries. Two vertices are adjacent if two countries have a
common boarder. So we reduce a problem to vertex
coloring.
Adjacent vertices must be colored in different colors.