The Bridge Obsession Problem

By

Vamshi Krishna Vedam

Topics to be discussed

Real-world Problem Formulating a problem to the Graph Properties Solution

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.

People staying there always wondered whether there was any way to walk

Over all the bridges exactly once.

Can you find a way?

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”

Formulating a Problem to the Graph

Consider

each area as a Vertex.

Each bridge as a edge.

Graph

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

verticesedges

Traversable Graphs

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

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.

PropertiesA 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

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.

Euler’s Solution for a ProblemEuler’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

a b

d c

e

a b

c d e

No.Of Odd Vertices

Traversable Comments

0 yes Start anywhere end back at the same point.

2 yes Start at one odd and end at another odd.

More than 2 No

Adding 8th Bridge

Becomes Euler Path

Adding 9th BridgeBecomes Euler Circuit

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.

References

1. http://scienceworld.wolfram.com/biography/Euler.html

2. http://www.jcu.edu/math/vignettes/bridges.htm

3. http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.html

4. http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg

5. http://en.wikipedia.org/wiki/Eulerian_path