Tuesday, January 17, 2012

Origin of Graph Theory

Graph Theory:

In this post, I would briefly go over through how and what led to the development of the graph theory which revolutionized the way many complicated problems were looked at and were solved.


The Königsberg bridge problem originated in the city of Königsberg, formerly in Germany but, now known as Kaliningrad and part of Russia, located on the river Preger. The city had seven bridges, which connected two islands with the main-land via seven bridges. People staying there always wondered whether was there any way to walk over all the bridges once and only once. The below picture is the map of Königsberg during Euler's time showing the actual layout of the seven bridges, highlighting the river Preger and the bridges.

Fig. 1

 In, 1736 Euler came out with the solution in terms of graph theory. He proved that it was not possible to walk through the seven bridges exactly one time. In coming to this conclusion, Euler formulated the problem in terms of graph theory. He abstracted the case of Königsberg by eliminating all unnecessary features. He drew a picture consisting of dots” that represented the landmasses and the line-segments representing the bridges that connected those land masses. The resulting picture might have looked somewhat similar to the figure shown below.

Fig. 2

This simplifies the problem to great extent. Now, the problem can be merely seen as the way of tracing the graph with a pencil without actually lifting it. One can try it in all possible ways, but you will soon figure out, it is not possible. Euler proposed that any given graph can be traversed with each edge traversed exactly once if and only if it had, zero or exactly two nodes with odd degrees. The graph following this condition is called, Eulerian circuit or path.

In actual case of seven bridges of Königsberg, once the situation was presented in terms of graph, the case was simplified as the graph had just 4 nodes, with each node having odd degree. So, Euler concluded that these bridges cannot be traversed exactly once.

Using this theorem, we can create and solve number of problems. Suppose now, we want to make the graph created from bridges of nigsberg, a Eulers circuit. Now, as per Euler’s theorem we need to introduce a path to make the degree of two nodes even. And other two nodes can be of odd degree out of which one has to be starting and other another the endpoint. Suppose we want to start our journey from blue node and end at the yellow node. So, the two nodes can have odd edges. But somehow we need to edit the actual graph by adding another edge to the graph such that the two other nodes have even degree. So, the resulting figure is shown below.

Fig. 3

No comments:

Post a Comment