Eulerizing Graph
Ch 5
Euler Circuit and Euler path
• When a graph has no vertices of odd degree,
then it has at least one Euler circuit.
• When a graph has exactly two vertices of odd
degree, then it has at least one Euler path.
• When a graph has more than two vertices of odd
degree, then
– There is no Euler circuit or Euler path
– There is no possible way that we can travel along all
the edges of the graph without having to recross
some of them.
Euler circuit & Euler path
• How we will find a route that covers all the edges
of the graph while revisiting the least possible
number of edges (deadhead travel)?
• In real-world problems,
The cost is proportional to the amount of travel.
• Total cost of a route
= cost of traveling original edges in the graph
+ cost of deadhead travel
Graph 1: No Euler circuit, No Euler path
A
B
C
L
D
E
K
Original graph
F
G
J
I
H
Represents vertex of odd degree
The graph has total 8 vertices of odd degree
=> The graph has no Euler circuit or Euler path
An eulerized version of the graph
A
B
C
D
L
E
K
F
G
J
I
H
By adding a duplicate copy of edges BC, EF, HI, and KL to
the previous graph, we get the above graph.
The graph has all vertices of even degree
=> The graph has Euler circuits
An eulerized version of the graph
A
B
1
28
9
18
L
27
K
17
26
J
10 C
2
19
8
16
25
3
7
I
D
11
12
20
13
4
15
6
24
5
14
23
E
21
F
22
G
H
The graph has all vertices of even degree
The graph has Euler circuits
Travel as numbered, you will get one such Euler circuit.
Trip along the edges of our original graph
A
28
B
1
9
18
L
27, 17
K
26
J
2, 10
19
25
7
I
D
11
3
8
16
C
12
20
4
15
6, 24
5
14
23
E
13, 21
F
22
G
H
In this trip we are traveling along all of the edges of the
graph, but we are retracing four of them: BC, EF, HI, and KL.
This is not an Euler circuit for the original graph, it is a circuit
describing the most efficient trip – an optimal circuit.
Eulerizing a graph
• We have started with a graph
• Modifying it by adding extra edges in such a way
that the odd vertices become even vertices.
• The process of changing a graph by adding
additional edges so that odd vertices are
eliminated is called eulerizing the graph.
• Note: The edges that we add must be duplicates
of edges that already exist. The idea is to cover
the edges of the original graph in the best
possible way without creating any new.
Graph 2: Another Example
A
B
C
D
E
P
F
O
G
N
H
Original graph
I
M
L
K
J
Represents vertex of odd degree
The graph has total 12 vertices of odd degree
=> The graph has no Euler circuit or Euler path
Illegal eularization of the graph
A
B
C
D
E
P
F
O
G
N
H
I
M
L
K
J
Edges DF and NL were not part of the original graph
Inefficient eularization of the graph
A
B
C
D
E
P
F
O
G
N
H
I
M
L
K
J
Optimal eularization of the graph
A
B
C
D
E
P
F
O
G
N
H
I
M
L
K
J
Semi- eularization of the graph
• We are not required to start and end at the
same vertex.
• We duplicate as many edges as needed to
eliminate all odd vertices except for two,
which we allow to remain odd.
• We then use one of the odd vertices as
the starting point and we will end up to the
other odd vertex.
Semi- eularization of the graph
A
B
C
D
E
P
F
O
G
N
H
Original graph
I
M
L
K
J
Represents vertex of odd degree
The graph has total 12 vertices of odd degree
=> The graph has no Euler circuit or Euler path
Semi- eularization of the graph
A
B
C
D
E
P
F
O
G
N
H
I
M
L
K
J
Vertices D and F remain odd
All other vertices are now even that were originally odd:
B, C, G, H, J, K, L, N, O, and P
Semi-eularization of the graph
• Semi-eularization tells us that is possible
to travel all the edges of the graph by
starting at D and ending at F (or vice
versa) and
• duplicating only six of the edges: BC, GH,
JK, LM, MN, and OP
• There are many other ways to accomplish
this, but none of them can do with fewer
than 6 duplicate edges.