2. Graphs consist of
points called vertices
lines called edges
1. Edges connect two
vertices.
2. Edges only intersect
at vertices.
3. Edges joining a
vertex to itself are
called loops.
The following picture is a graph.
A
B
C
D
E
3. EULER GRAPH
• A graph is called Eulerian if it has an Eulerian
Cycle and called Semi-Eulerian if it has an
Eulerian Path.
An Eulerian cycle (path) is a sub_graph
Ge = (V;Ee) of G = (V;E)
which passes exactly once through each edge
of G.
G must thus be connected and all vertices V are
visited (perhaps more than once). We then says
that G is Eulerian
4. 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 cross some of them
more than once.
5. 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
6. The bridges of Konigsberg problem
Seven bridges of Konigsberg problem, was first solved by Euler in
the eighteenth century. The problem was rather simple. The town of
Konigsberg consists of two islands and seven bridges. Is it possible,
by beginning anywhere and ending anywhere, to walk through the
town by crossing all seven bridges but not crossing any bridge twice?
7. Eulerizing a graph
• Modifying a graph 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.
8. Eulerizing Graphs
First step is to identify the
odd vertices.
Second step is to add
duplicate copies of edges to
create all even vertices
OPTIMAL
ROUTE:
duplicate the
fewest
number of
edges
NOT an
optimal
route
illegal
route
9. 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.
10. Semi-Eulerizing Graphs
First step is to identify the odd
vertices.
Second step is leave out 2 odd
vertices and add duplicate copies
of edges to create even vertices
OPTIMAL
ROUTE:
duplicate the
fewest
number of
edges
NOT an
optimal
route
illegal
route
11. Fleury’s Algorithm
• Finds an Euler circuit in a connected graph
with no odd vertices.
• Finds an Euler path in a connected graph with
two odd vertices.
• The idea behind the algorithm: Don’t burn
your bridges behind you.
A B
Bridge
We would only want to cross that bridge if we know
that all edges in A have been traveled.
12. Fleury’s Algorithm
• Let G be an Eulerian graph.
• STEP 1: Choose any vertex v of G and set current vertex equal to v and
current trail equal to the empty sequence of edges.
• STEP 2: Select any edge e incident with the current vertex but choosing a
bridge only if there is no alternative.
• STEP 3: Add e to the current trail and set the current vertex equal to the
vertex at the ‘other end’ of e. [If e is a loop, the current vertex will not
move.]
• STEP 4: Delete e from the graph. Delete any isolated vertices.
• Repeat steps 2 – 4 until all edges have been deleted from G. The final
current trail is an Eulerian trail in G.
13. Fleury’s Theorem
• Every time we traverse another edge, we erase it from
copy 1 but mark (red) and level it with the
appropriate number on copy 2.
• Copy 1 gets smaller and cop2 gets redder.
• Copy 1 helps us decide where to go next; copy 2
helps us reconstruct our trip.
Copy 1
A
B
C D
E F
B
C D
E F
Copy 2
Start at F (arbitrarily)
14. Fleury’s Theorem
Copy 1
B
C D
E F
A
B
C D
E F
Copy 2
Step 1: Travel from F to C
Could have also travel
from F to D
A
15. Fleury’s Theorem
Copy 1
B
C D
E F
A
B
C D
E F
Copy 2
Step 2: Travel from C to D
Could have also travel
to A or to E
1
2
A
16. Fleury’s Theorem
Copy 1
B
C D
E F
A
B
C D
E F
Copy 2
Step 3: Travel from D to A
Could have also travel
to B but not to F – DF is a bridge!
1
2
A
3
17. Fleury’s Theorem
Copy 1
B
C D
E F
A
B
C D
E F
Copy 2
Step 4: Travel from A to C
Could have also travel
to E but not to B – AB is a bridge!
1
2
A
3
4
18. Fleury’s Theorem
Copy 1
B
C D
E F
A
B
C D
E F
Copy 2
Step 5: Travel from C to E
There is no choice!
1
2
A
3
4
5
19. Fleury’s Theorem
Copy 1
B
C D
E F
Copy 2
Steps 6, 7, 8, and 9: Only one way to go at each step
1
2
A
3
4
5
6
7
8
9
20. Fleury’s Algorithm for finding Euler
circuit
• First make sure that the graph is connected and all
the vertices have even degree.
• Pick any vertex as the stating point
• When you have a choice, always choose to travel
along an edge that is not a bridge of the yet-to-be-
traveled part of the graph
• Label the edges in the order in which we travel them
• When we cannot travel any more, stop. we are done.
