Computer Science
Design and Analysis of
Algorithms
2D Animation using JAVA
To find out whether the graph has Hamiltonian cycle if there exists a
Hamiltonian path.
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Hamiltonian Path (HP)
A Hamiltonian path is a path in an undirected
graph which visits each vertex exactly once.
Hamiltonian cycle (HC)
A Hamiltonian cycle (or Hamiltonian circuit) is a
cycle in an undirected graph which visits each
vertex exactly once and also returns to the starting
vertex.
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Aim and procedure
 Aim is to show this algorithm by animation so that it become
easy to visualize what is going on with removing edges and
vertices.
 Procedure:
• If it doesn’t contains HP:
If it doesn’t have Hamiltonian Path then it will definitely won’t
have HC. As every HC has HP in it.
•
If it contains HP :
In this case graph may have HC or may not. For that we need to
remove edge and new vertices in the graph.
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Problem Statement
Input:
 A graph
 Yes(If graph has HP)
 No(If graph doesn’t have HP)
Output:
 Yes(If the graph has HC)
 No(If the graph doesn’t have HC)
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--GRAPH
--Red line represents HP in
the Graph
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Case 1: There is no HP in the Graph
Path 1
Path 2
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We can see that if the graph doesn’t have HP in it then it
cannot have HC in it in any case.
HP is a subset-path of HC and if there is no HP than their cannot be HC . We saw this
by the above example.
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Case 2: If there exists an HP in the graph
Procedure : Remove one edge and join the two vertices
with two newly created vertices.
Now check whether that new graph has HP
or not. If it has HP than the original graph has
HC in it.
Note : We have to do this for all the edges if altering any
edge we find HP than we have HC in the original graph.
But if we don’t find HP after trying with all edges than only
the previous graph don’t have HC in it.
If the Graph has HP it may have HC or not depends on the above procedure.
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Example for existence of HC while there is HP
First choosing this edge
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Starting from this point we will try to find a
HP in it.
Take any two vertices and join them with two new vertices
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Path 1
Got stuck from this path
And was not able to get
HP this way.
Path 2
We find out a HP in the new graph via path 2.
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Since we are able to find a HP in the new graph
therefore there exist a HC in the original graph
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Example for no existence of HC while there is HP
There is a HP in the graph but we’ll
see that there is no HC this time.
Take any two vertices and remove the edge
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Join them with two new vertices
We are going to search for
HP in the new graph through
this point.
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Option 1
Option 2
Path 2
Path 1
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Path 1
This point has now 2 options.
No HP through any further path
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Path 2
Similarly here we have 3 options and similarly doing the above steps to find HP we
didn’t reach HP.
Blue green and Red lines show the different
option we could follow to find HP but none of
them leads to HP
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We removed this edge but here we were
Not able to find HP in the new graph.
Similarly we need to check for all the edges and if we are not able to find HP through any case
Then we will conclude that the original graph doesn’t have HC
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Trying to find HP on
removing this edge now.
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Starting from this point
Similarly as previous example after trying every possible path we didn’t reach HP.
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We tried for the above two edges, similarly trying for every edge we were not able to find HP
Conclusion: After checking with all the edges, we find there doesn’t exists HP therefore the
Original graph doesn’t contain HC in it.
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Review Questions
1.
2.
3.
4.
If there is HP in a graph than does it necessarily have HC in it. (Answer – No)
If there is no HP in a graph than does it necessarily don’t have HC in it. (Answer – Yes)
Do HC covers all vertices. ( Answer – Yes)
Do black lines in the shown graph represents HC.(Answer – No, because in making of that
path you need to pass the centre vertex
twice, which is not allowed in HC)
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CREDITS
Saurabh Chanderiya
07005004
Computer Science and Engineering.
[email protected]
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