Graph Theory Two Applications D.N. Seppala-Holtzman St. Joseph ’ s College

Here we will consider two applications of graph theory Euler Circuits Planarity

The 7 Bridges of Konigsburg Konigsburg (now called Kalingrad) is a city on the Baltic Sea wedged between Poland and Lithuania. A river runs through the city which contains a small island. There are 7 bridges which connect the various land masses of the city.

The City of Konigsburg

The Problem The people of Konigsburg made a sport during the 18 th century of trying to cross each and every one of the 7 bridges exactly once. This was to be done in such a way that one would always end up where one began.

The problem was solved by Leonhard Euler ( )

Euler (pronounced “oiler”) Euler was one of the greatest mathematicians of all time. He contributed to virtually every field of mathematics that existed in his time. The publication of his collected works (Opera Omnia) is presently up to volume 73 and still not complete.

Euler and Graph Theory Euler’s solution to the Konigsburg bridge problem was more than a trivial matter. He didn’t just solve the problem as stated; he made a major contribution to graph theory. Indeed, he essentially invented the subject. His contribution has many practical applications.

Some Vocabulary A graph is a set of vertices connected by edges. The valence of a vertex is the number of edges that meet there. An Euler Circuit is a path within a graph that covers each and every edge exactly once and returns to its starting point.

Euler’s Theorem A connected graph has an Euler circuit if and only if every vertex has an even valence. The Konigsburg bridge problem translated into a graph in which all valences were odd. Thus there was no way to walk on each bridge precisely once.

Euler’s Theorem --- Why is it true? Any vertex with odd valence must be either a starting point or an ending point. All points that are neither starting nor ending points must be left as often as they are entered.

Euler’s Theorem --- Why is it important? There are many, many examples of circuits that one wishes to traverse such that every edge is covered and no edge is repeated. Routes for snowplows, letter carriers, meter readers, and the like, share these characeristics.

Not all graphs have even valence on all vertices --- What then? One cannot expect that every street layout or route will translate into a graph with all vertices of even valence. In these cases, one can try to minimize the number of edges that are repeated. There is an algorithm to do this. It is called Eulerizing the graph.

Eulerizing a Graph I Select pairs of vertices in the graph that have odd valence. Do this in such a way that the vertices are as close together (have the fewest edges between them) as possible. Neighboring vertices would be the best choice, if possible.

Eulerizing a Graph II For each edge on the path that connects a pair of odd-valenced vertices, generate a “phantom edge” duplicating that edge. Do this for each pair of odd-valenced vertices. In general, there will be more than one Eulerization of a graph. The fewer duplicated edges, the better.

Recall the City of Konigsburg

Let us Eulerize Konigsburg I A B C D

Let us Eulerize Konigsburg II A B C D

Eulerizing Konigsburg III Here, we have selected pairs of odd-valenced vertices, BD and AC. We have added a “phantom” edge between these pairs of vertices. These phantom edges are edges that are traversed twice. Now, with the addition of just two edges, the graph has all even-valenced vertices.

A Troublesome Question How do we know that we can always do this? In particular, how do we know that the odd-valenced vertices will occur in pairs?

The Number of Odd-Valenced Vertices is Even ---Here’s a Proof: Suppose that there are N edges. Thus, there are 2N “ends” of edges. The sum of all the valences must be 2N. Thus, it is not possible to have an odd number of odd-valenced vertices. Hence, the odd-valence vertices occur in pairs.

Euler Circuits: In Summation A very simple and elegant idea has led to a wide variety of real-world applications. Nearly any process which involves routing (and there are many) can be made more efficient by these methods. Many millions of dollars can be saved in the process!!

Planarity Another problem in graph theory also has a simple solution that has major consequences. The question of planarity refers to whether a graph can be drawn in the plane without any edges crossing any other ones.

Connect 3 Houses to 3 Utilities H1H2H3 U1U2U3 Draw edges from each U to each H without crossing edges.

An Attempted Solution H1H2H3 U1U2U3 No H2-U2 Connector

K 3,3 The graph connecting all vertices of a set of three to all vertices to another set of three is called K 3,3 This graph is not planar. That is to say, it is not possible draw it in the plane with no edges crossing others.

KnKn K n is called the complete graph on n vertices. It is the graph one gets by starting with n vertices and drawing an edge between each pair. K n is planar or not depending upon n.

K 3 Is Just a Triangle, Thus It Is Planar

K 4 Is Also Planar

K 5 Is Not Planar A B C D E No E B Edge

K n Is Not Planar for n > 4 We know that K 5 is not planar. If n is bigger than or equal to 5 then K n couldn’t possibly be planar.

Planar Graphs --- A Theorem All non-planar graphs (those that cannot be drawn in the plane without crossing edges) contain either a copy of K 5 or K 3,3 as a sub-graph. Conversely, if neither K 5 nor K 3,3 is to be found embedded anywhere inside a graph, that graph will be planar.

Planar Graphs --- Who Cares? Whether a graph is planar or not is quite important. Any physical interpretation of a graph that wants to avoid crossings of edges needs to take this into account. The most obvious examples are printed circuit boards and micro-chips.

Other Graph Theory Applications Euler Circuits and Planarity are only two of many applications of graph theory. Most any system or object which contains a network of some type has an application of graph theory lurking somewhere below the surface. Roads, routes, phone lines, electrical circuits are all examples.