1. The Bridge Obsession Problem By Vamshi Krishna Vedam

2. Topics to be discussed  Real-world Problem  Formulating a problem to the Graph  Properties  Solution

3. Konigsberg Bridge Problem:  The Bridge Obsession Problem is also called as The Seven Bridges Of Konigsberg problem.  The Konigsberg bridge problem originated in the city of Konigsberg, formerly in Germany but, now known as Kaliningrad and part of Russia, located on the river Pregel.  The city had seven bridges, which connected two islands with the main- land via seven bridges.

4.  People staying there always wondered whether there was any way to walk Over all the bridges exactly once.  Can you find a way?

5. Leonhard Euler (1707 - 1783)  Was a pioneering Swiss Mathematician and Physicist.  He made an important discoveries in fields as diverse as infinitesimal calculus and Graph Theory.  Invented the notation i, π, e, sin, cos, f(x) and more!  Lost sight in both eyes but became more productive, saying “now I have fewer distractions”

6. Formulating a Problem to the Graph Consider  each area as a Vertex.  Each bridge as a edge.

7. Graph A Graph is a set of points called vertices or nodes, connected by lines called edges. vertices edges

8. Traversable Graphs Which of these graphs can be drawn without taking your pen off the paper or repeating any edges?

9. Euler Path and Circuit  An Euler path in a graph is a path that uses every edge of the graph exactly once.  An Euler circuit in a graph is a circuit that uses every edge of the graph exactly once.

10. Properties A graph has an Euler path if and only if it is connected and exactly two of its vertices have odd degrees. A graph has an Euler circuit if and only if it is connected and all its vertices have even degrees

11. Degree Of a Vertex Number of Graph edges meeting at a given node. A vertex with an odd number of edges leads to Odd Vertex. A vertex with an even number of edges leads to Even Vertex.

12. Euler’s Solution for a Problem Euler’s Theorem  If a network has more than two odd vertices, it does not have an Euler path.  If a network has two or less odd vertices, it has at least one Euler path. Which Of the following are Euler Paths? a d b e c ab d c e ab cde

13. No.Of Odd Vertices TraversableComments 0yesStart anywhere end back at the same point. 2yesStart at one odd and end at another odd. More than 2No

14. Adding 8 th Bridge Becomes Euler Path

15. Adding 9 th Bridge Becomes Euler Circuit

16. Solution: An Impossible Problem If we look again at the map of Konigsberg, we see that there are an odd number of bridges coming out of every bit of land, so such a walk around the city is impossible.

17. References 1.http://scienceworld.wolfram.com/biography/Euler.htmlhttp://scienceworld.wolfram.com/biography/Euler.html 2.http://www.jcu.edu/math/vignettes/bridges.htmhttp://www.jcu.edu/math/vignettes/bridges.htm 3.http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.html 4.http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberghttp://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg 5.http://en.wikipedia.org/wiki/Eulerian_pathhttp://en.wikipedia.org/wiki/Eulerian_path