Euler Paths & Circuits
Hamilton Paths & Circuits
Euler Paths
and Euler Circuits
Review from last lesson
▪ adjacent vertices – vertices that are connected directly and thus share at least one edge
▪ path – a sequence of adjacent vertices and the edges
connecting them, denoted by a list of vertices in order
▪ circuit – a path that begins and ends at the same vertex
path: A,B,F,G,H,M
circuit: A,B,F,G,L,K,J,E,A
NOTE 1: An edge can be part of a path only once.
Definitions
▪ Euler path – a path that travels through every edge of a graph once and only once.
▪ Euler circuit – a circuit that travels through every edge of a graph once and only once.
OR – a Euler path that begins and ends at the same vertex.
So, if the graph is traversable then it has a Euler path!
Every Euler circuit is an Euler path
Euler’s Rules of Traversability
1. A graph with all even vertices is traversable.
One can start at any vertex and end at same vertex.
2. A graph with two odd vertices is traversable.
One must start at one odd vertex and
end at the other odd vertex.
3. A graph with more than two odd vertices is
NOT traversable.
NOTE: Rules are only for
connected
graphs.
Euler
Circuit
Euler
Path
Example
Find the Euler path or circuit, if any.
Even (2)
Even (4)
Odd (3)
Even (4)
Odd (3)
2. A graph with two odd vertices is traversable. One must start at one odd vertex and end at the other odd vertex.
Example
Find the Euler path or circuit, if any.
2. A graph with two odd vertices is traversable. One must start at one odd vertex and end at the other odd vertex.
Euler Path
A
B
C
E
D
Euler Path: C,B,A,E,B,D,E,C,D
Euler Path: C,B,A,E,B,D,E,C
Euler Path: C,B,A,E,B,D
Euler Path: C,B,A,E,B
Euler Path: C,B,A,E
Euler Path: C,B,A
Euler Path: C,B
Euler Path: C
Euler Path:
Example
Find the Euler path or circuit, if any.
2. A graph with two odd vertices is traversable. One must start at one odd vertex and end at the other odd vertex.
Euler Path
3
2
4
1
6
7
5
8
Example
Find the Euler path or circuit, if any.
Even (4)
Even (2)
Even (4)
Even (2)
Even (2)
Even
(4)
Even (2)
1. A graph with all even vertices is traversable. One can start at anyvertex and end at same vertex.
Example
Find the Euler path or circuit, if any.
A
B
C
E
D
F
G
Euler Circuit:
Euler Circuit: A
Euler Circuit: A,B
Euler Circuit: A,B,C
Euler Circuit: A,B,C,G
Euler Circuit: A,B,C,G,E
Euler Circuit: A,B,C,G,E,C
Euler Circuit: A,B,C,G,E,C,D
Euler Circuit: A,B,C,G,E,C,D,E
Euler Circuit: A,B,C,G,E,C,D,E,F
Euler Circuit: A,B,C,G,E,C,D,E,F,G
Euler Circuit: A,B,C,G,E,C,D,E,F,G,A
1. A graph with all even vertices is traversable. One can start at any vertex and end at same vertex.
Example
Find the Euler path or circuit, if any.
1
2
6
8
7
9
3
Number the Euler Circuit at each step.
1. A graph with all even vertices is traversable. One can start at any vertex and end at same vertex.
Euler Circuit
5
Example
Find the Euler path or circuit, if any.
Even
(4)
Odd (3)
Odd (3)
Odd (3)
Odd (3)
3. A graph with more than two odd vertices is NOT traversable.
Example
Find the Euler path or circuit, if any.
Euler Circuit
1
10
7
14
5
6
2
4
13
12
9
8
Number the Euler Circuit at each step.
Example
Find the Euler path or circuit, if any.
Euler Path: B, L, D, O, L, O, D, K, O, B, K
▪ connected graph – a graph in which there is at least one path connecting any two vertices
▪ disconnected graph – a graph in which there is no path connecting any two vertices
▪ bridge – an edge that, if removed, would make a connected graph into a disconnected graph
disconnected graph
edge BG is a bridge
How do we find the
Euler path or Euler circuit?
Fleury’s Algorithm
▪ Check that the graph is connected.
▪ Check that the graph is traversable using Euler’s Rules.
▪ Choose a starting point based on Euler’s Rules.
▪ After each edge is traveled over, erase it to create a reduced graph. You may want to show the erased edge as a dotted line.
▪ When you have a choice between two edges, never take the bridge of a reduced graph. Travel over a bridge only when there is no other
alternative.
▪ Continue until you get to the appropriate vertex and the entire graph has been traversed.
Example
Find the Euler path or circuit, if any.
Is the graph connected? YES
Is the graph traversable? YES
Where do we start? ANYWHERE
If you have the choice between two edges, never take the bridge.
Neither is a bridge. Go A to D
Erase (make dotted) and number the edge that is traveled over. # 1
1
If you have the choice between two edges, never take the bridge.
None are a bridge. Go D to C.
Erase (make dotted) and number the edge that is traveled over. # 2
2
If you have the choice between two edges, never take the bridge.
CA is bridge – don’t cross it. The others are not bridges. Go C to F.
Erase (make dotted) and number the edge that is traveled over. # 3
3
If you have the choice between two edges, never take the bridge.
FE is bridge – don’t cross it. The others are not bridges. Go F to D.
Erase (make dotted) and number the edge that is traveled over. # 4
4
If you have the choice between two edges, never take the bridge.
There is no choice. Go D to B.
5
Erase (make dotted) and number the edge that is traveled over. # 5
If you have the choice between two edges, never take the bridge.
There is no choice. Go B to F.
Erase (make dotted) and number the edge that is traveled over. # 6
6
If you have the choice between two edges, never take the bridge.
There is no choice. Go F to E.
Erase (make dotted) and number the edge that is traveled over. # 7
7
If you have the choice between two edges, never take the bridge.
There is no choice. Go E to C.
Erase (make dotted) and number the edge that is traveled over. # 8
8
If you have the choice between two edges, never take the bridge.
There is no choice. Go C to A.
Erase (make dotted) and number the edge that is traveled over. # 9
9
The entire graph has now been traversed. DONE!
If you have
more than one choice that is not a bridge,
take either.
Hamilton Paths
and Hamilton Circuits
Euler paths and circuits cover every edge of a graph.
These are useful in optimizing routes for applications
like garbage collection, where each street (edge) only
needs to be traversed once but a particular intersection
(vertex) may be crossed more than once.
What about optimizing routes for applications like
FedEx or UPS in package delivery? For these
Definitions
▪ Hamilton path – a path that travels through every vertex of a graph once and only once.
▪ Hamilton circuit – a Hamilton path that begins and ends at the same vertex and passes through all other vertices exactly once.
This is not the same as being traversable.
In fact, every edge does not even have to be crossed.
Example
Examine the graph below.
3. A graph with more than two
odd vertices is NOT traversable.
Is there a Euler path or circuit?
No, there is neither.
Is there a Hamilton path or
circuit? Let’s try…
Yes, there is a Hamilton path.
and a Hamilton circuit.
Just as with the Euler type, if there is a Hamilton circuit there must be a
Example
Examine the graph below.
Is there a Hamilton path or
circuit? Let’s try…
Yes, there is a Hamilton path.
but NO Hamilton circuit.
Notice that all the vertices are
even. Thus the graph has an
Definition
▪ Complete graph – a graph that has an edge between each pair of its vertices
NOTE: This is not the same as a connected graph. In a connected graph, all the vertices connect through some path which may travel over several edges. In a complete graph, there is a direct line, or edge, between each pair of vertices.
Complete Graph Rule
Every complete graph with 3 or more vertices has a Hamilton circuit.
Complete Graph Rule
Every complete graph with 3 or more vertices has a Hamilton circuit. (Thus it also has a Hamilton path.)
Complete Graph Rule
Every complete graph with 3 or more vertices has a Hamilton circuit. (Thus it also has a Hamilton path.)
Example
Example
These are complete graphs:
Example
Find a Hamilton circuit:
A
B
C
D
E
One possibility: A, B, C, D, E, A
What about finding a different one?
B, C, D, E, A, B is the SAME CIRCUIT!
B, C, D, E, A, B ?
There are actually (
n
-1)! circuits in a complete graph with
n
vertices. To avoid duplication when listing them, the book
always starts with “A” only.
Example
Find a Hamilton circuit:
A
B
C
D
E
One possibility: A, B, C, D, E, A
What about finding a different one?
A, C, E, B, D, A
Definition
▪ Weighted graph – a complete graph whose edges have numbers, or weights, attached to them
Example
A sales director lives in City A and must fly to the regional
offices in B, C, and D. There are direct flights between
each pair of cities. He will return home at the end of the
business trip. The chart below shows the airfares for all
possible flights.
A B C D
A * $ 190 $ 124 $ 157
B $ 190 * $ 126 $ 155
C $ 124 $ 126 * $ 179
D $ 157 $ 155 $ 179 *
COMPLETE GRAPH HAMILTON
Example
A sales director lives in City A and must fly to the regional
offices in B, C, and D. There are direct flights between
each pair of cities. He will return home at the end of the
business trip. The chart below shows the airfares for all
possible flights.
A B C D
A * $ 190 $ 124 $ 157
B $ 190 * $ 126 $ 155
C $ 124 $ 126 * $ 179
D $ 157 $ 155 $ 179 *
A
B
C
D
How can the visits be scheduled
in the cheapest possible way?
Need a weighted
Example
A sales director lives in City A and must fly to the regional
offices in B, C, and D. There are direct flights between
each pair of cities. He will return home at the end of the
business trip. The chart below shows the airfares for all
possible flights.
A B C D
A * $ 190 $ 124 $ 157
B $ 190 * $ 126 $ 155
C $ 124 $ 126 * $ 179
D $ 157 $ 155 $ 179 *
A
B
C
D
How can the visits be scheduled
in the cheapest possible way?
190
124
157
126
Example
A sales director lives in City A and must fly to the regional
offices in B, C, and D. There are direct flights between
each pair of cities. He will return home at the end of the
business trip. The chart below shows the airfares for all
possible flights.
A
B
C
D
How can the visits be scheduled
in the cheapest possible way?
190
124
157
126
155
179
List all circuits.
A
B
C
D
How can the visits be scheduled
in the cheapest possible way?
190
124
157
126
155
179
List all circuits.A, B, C, D, A
A, B, D, C, A
A, C, B, D, A
A, C, D, B, A
A, D, B, C, A
Example
A, D, C, B, A
Find total weights.
= 190
= 190 + 155 + 179 + 124 = 648
= 124 + 126 + 155 + 157 = 562
= 124 + 179 + 155 + 190 = 648
= 157 + 155 + 126 + 124 = 562
= 157 + 179 + 126 + 190 = 652
For $ 562 can travel either:
A, C, B, D, A
A, D, B, C, A
Notice these are reversed.
Brute Force Method… calculate every possibility and see which is best.
A
B
C
D
190
124
157
126
155
179
Example
Another way to find the correct route…
Nearest Neighbor Method
From the starting point, choose the edge with the smallest weight.
Continue choosing the edge with the smallest weight without going
back to a previous vertex.
A can move to B, C, or D.
Which has smallest weight?
C
124
C can move to B or D.
Which has smallest weight?
B
+ 126
B can only move to D.
D
+ 155
D can only return to A.
A
+ 157 = 562
NOTE: The Nearest Neighbor Method only approximates the smallest
Example
Use the Nearest Neighbor Method to approximate
the solution to the weighted graph below.
A
B
C
D
E
128
114
180
147
195
145
116
169
194
115
A
C
E
D
B
A
114
+ 115
+ 194 + 145 + 180
= 748
Optimal solution is A, E, C, B, D, A (or its reverse) for
Homework
From the Cow book 10.7 pg 549 # 1 – 6 all 15.1 pg 786 # 1 – 47 odd
15.2 pg 796 # 1 – 39 odd
