Route Inspection
Chinese Postman Problem
Objectives
4.1 Determine whether a graph is traversable
4.2 Use the route inspection (Chinese postman
algorithm to find the shortest inspection
cycle in a network).
4.1 Determine whether a graph is traversable
The degree or valency or order of a vertex is the number of
arcs incident to it.
2.2 Basic terminology used in graph theory
(orof
nodes)
If •theVertices
degree
a vertex is even, we
• itVertex
set degree, so J, K and M
say
has even
• Edges
arcs)
have
even(or
degree.
• Edge set
Similarly
vertices L and N have odd
• Subgraph
degree.
• Degree (or valency, or order of a vertex)
• Even degree
• Odd degree
In •anyPath
graph the sum of the degrees will
Walk equal to 2 x the number of
be• precisely
edges.
This(oris circuit)
because, in finding the sum
• Cycle
of•degrees,
we are counting each end of
connected
each edge.
Vertex
Degree
J
2
K
2
1
4
1
• Not connected
L
• loop
M
• Simple graph
N
This is known as the Handshaking Lemma.
• Digraph
• Directed edges
4.1 Determine whether a graph is traversable
How many valencies are there for
each vertex?
2
4
4
4
4
4
If all the valencies in a graph are even,
then the graph is Eulerian.
Is this graph traversable?
2
A graph is traversable if it is possible
to traverse (travel along) every arc
just once without taking your pen
from the paper.
4.1 Determine whether a graph is traversable
2
1
How many valencies are there for
each vertex?
2
4
4
2
2
1
If precisely two valencies in a graph
are odd, and the rest are even, then
the graph is semi-Eulerian.
Is this graph traversable?
A graph is traversable if it is possible
to traverse (travel along) every arc
just once without taking your pen
from the paper.
4.1 Determine whether a graph is traversable
Which of the following graphs are traversable?
2
4
4
4
2
3
1
√
2
1
3
2
2
1
3
1
2
×
3
3
4
1
3
1
3
×
1
4
3
3
4
A graph is traversable if all the valencies are even.
A graph is semi-traversable if it has precisely two odd valencies.
A graph is not traversable if it has more than two odd valencies.
4
√
2
4.1 Determine whether a graph is traversable
Which of the following graphs are traversable?
2
4
4
4
2
3
1
√
2
1
3
2
2
1
3
1
2
×
3
3
4
1
3
1
3
×
1
4
3
3
4
4
2
√
semi-Eulerian
Eulerian
All Eulerian graphs are traversable.
A graph is traversable if all the valencies are even.
All
semi-Eulerian graphs are semi-traversable.
A graph
is semi-traversable
if itthe
hasfinish
precisely
valencies.
In
this case
the start point and
pointtwo
willodd
be the
two vertices with
A graph
is not traversable if it has more than two odd valencies.
odd
valencies.
4.1 Determine whether a graph is traversable
1. a. Verify that the graph is Eulerian.
b. Find a route, starting and finishing at A, that traverses the graph.
a.
Vertex
A
B
C
D
E
F
G
Valency
2
4
4
2
4
4
4
All valencies are even, so the graph is Eulerian.
b. A possible route is:
A, B, C, D, E, C, G, F, E, G, B, F, A.
4.1 Determine whether a graph is traversable
2. Find a route that traverses each arc of this graph just once.
You may start and finish at different points.
3
3
This graph has precisely two odd
valencies, so it is semi-Eulerian.
A possible route is
C
2
D
2
If a graph is semi-traversable,
then the start point and the
finish point will be the two
vertices with odd valencies.
A, B, D, C, A, B.
4.1 Determine whether a graph is traversable
3. Prove that there must always be an even (or zero) number of
vertices with odd valency in every graph.
2.2 Basic terminology used in graph theory
(orof
nodes)
If •theVertices
degree
a vertex is even, we
• itVertex
set degree, so J, K and M
say
has even
• Edges
arcs)
have
even(or
degree.
• Edge set
Similarly
vertices L and N have odd
• Subgraph
degree.
• Degree (or valency, or order of a vertex)
• Even degree
• Odd degree
In •anyPath
graph the sum of the degrees will
Walk equal to 2 x the number of
be• precisely
edges.
This(oris circuit)
because, in finding the sum
• Cycle
of•degrees,
we are counting each end of
connected
each edge.
Vertex
Degree
J
2
K
2
1
4
1
• Not connected
L
• loop
M
• Simple graph
N
This is known as the Handshaking Lemma.
• Digraph
• Directed edges
4.1 Determine whether a graph is traversable
3. Prove that there must always be an even (or zero) number of
vertices with odd valency in every graph.
Each arc has two ends
and so will contribute two to the sum of the valencies of the
whole graph.
The sum of the valencies = Number of arcs × 2
The sum of the valencies is even.
Any odd numbers must occur in pairs.
There is an even number of odd valencies.