the seven bridges of konigsberg-euler's solutiondragan/st-spring2016/the seven bridges of...
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The Seven Bridges of
Konigsberg-Euler's solution
Ajitesh vennamaneni
810838689
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Content
Real world problem
Graph construction
Special properties
Solution
applications
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The Seven Bridges of Konigsberg
• The problem goes back to year 1736.
• This problem lead to the foundation of graph theory.
• In Konigsberg, a river ran through the city such that in its
center was an island, and after passing the island, the
river broke into two parts.
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R-W Problem
• We have seven bridges
for people of the city to
get from one part to
another
• The people wondered
whether or not one could
walk around the city in a
way that would involve
crossing each bridge
exactly once.
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Try solving it
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Few tries
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Lets construct a graph from that R-W
problem
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Odd and even vertex
Odd vertex
A vertex is called odd if it has an
odd number of edges leading to
it.
Even vertex
A vertex is called even if it has an
even number of edges leading to
it.
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Special Euler's properties
To get the Euler path a graph should have two or less number of odd
vertices.
Starting and ending point on the graph is a odd vertex.
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Problem faced
A vertex needs minimum of two
edges to get in and out.
If a vertex has odd edges then
the person gets trapped.
Hence every odd vertex should
be a starting or ending point in
the graph.
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In our problem graph we have
four odd vertices hence there
cant be any Euler path possible.
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solution
A bridge is added between C
and D
This makes the number of odd
vertices 2 and number of even
vertices 2 which satisfies our
properties
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Other Possible Solutions
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Applications
Transportation
Biology
Chip designing
Chemistry ….
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Reference
http://mathforum.org/isaac/problems/bridges1.html
http://www.maa.org/press/periodicals/convergence/leonard-eulers-
solution-to-the-konigsberg-bridge-problem-konigsberg
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Any Queries ??