4.1: Graphs
Back in the 18 th century in the Prussian city of Königsberg, a river ran through the city and seven bridges crossed the forks of the river. The river and the bridges are highlighted in the picture to the right[1].
As a weekend amusement, townsfolk would see if they could find a route that would take them across every bridge once and return them to where they started.
Solution
Leonard Euler (pronounced OY-lur), one of the most prolific mathematicians ever, looked at this problem in 1735, laying the foundation for graph theory as a field in mathematics. To analyze this problem, Euler introduced edges representing the bridges:
Since the size of each land mass it is not relevant to the question of bridge crossings, each can be shrunk down to a vertex representing the location:
Notice that in this graph there are two edges connecting the north bank and island, corresponding to the two bridges in the original drawing. Depending upon the interpretation of edges and vertices appropriate to a scenario, it is entirely possible and reasonable to have more than one edge connecting two vertices.
While we haven’t answered the actual question yet of whether or not there is a route which crosses every bridge once and returns to the starting location, the graph provides the foundation for exploring this question.
Definitions
While we loosely defined some terminology earlier, we now will try to be more specific.
A vertex is a dot in the graph that could represent an intersection of streets, a land mass, or a general location, like “work” or “school”. Vertices are often connected by edges. Note that vertices only occur when a dot is explicitly placed, not whenever two edges cross. Imagine a freeway overpass – the freeway and side street cross, but it is not possible to change from the side street to the freeway at that point, so there is no intersection and no vertex would be placed.
Edges connect pairs of vertices. An edge can represent a physical connection between locations, like a street, or simply that a route connecting the two locations exists, like an airline flight.
A loop is a special type of edge that connects a vertex to itself. Loops are not used much in street network graphs.
The degree of a vertex is the number of edges meeting at that vertex. It is possible for a vertex to have a degree of zero or larger.
A path is a sequence of vertices using the edges. Usually we are interested in a path between two vertices. For example, a path from vertex A to vertex M is shown below. It is one of many possible paths in this graph.
A circuit is a path that begins and ends at the same vertex. A circuit starting and ending at vertex A is shown below.
A graph is connected if there is a path from any vertex to any other vertex. Every graph drawn so far has been connected. The graph below is disconnected ; there is no way to get from the vertices on the left to the vertices on the right.
Here is a portion of a housing development from Missoula, Montana[2]. As part of her job, the development’s lawn inspector has to walk down every street in the development making sure homeowners’ landscaping conforms to the community requirements.
Solution
Naturally, she wants to minimize the amount of walking she has to do. Is it possible for her to walk down every street in this development without having to do any backtracking? While you might be able to answer that question just by looking at the picture for a while, it would be ideal to be able to answer the question for any picture regardless of its complexity.
To do that, we first need to simplify the picture into a form that is easier to work with. We can do that by drawing a simple line for each street. Where streets intersect, we will place a dot.
This type of simplified picture is called a graph .
While we drew our original graph to correspond with the picture we had, there is nothing particularly important about the layout when we analyze a graph. Both of the graphs below are equivalent to the one drawn above since they show the same edge connections between the same vertices as the original graph.
In the first section, we created a graph of the Königsberg bridges and asked whether it was possible to walk across every bridge once. Because Euler first studied this question, these types of paths are named after him.
An Euler path is a path that uses every edge in a graph with no repeats. Being a path, it does not have to return to the starting vertex.
In the graph shown below, there are several Euler paths.
Solution
One such path is CABDCB. The path is shown in arrows to the right, with the order of edges numbered.
An Euler circuit is a circuit that uses every edge in a graph with no repeats. Being a circuit, it must start and end at the same vertex.
The graph below has several possible Euler circuits.
Solution
Here’s a couple, starting and ending at vertex A: ADEACEFCBA and AECABCFEDA. The second is shown in arrows.
Look back at the example used for Euler paths – does that graph have an Euler circuit? A few tries will tell you no; that graph does not have an Euler circuit. When we were working with shortest paths, we were interested in the optimal path. With Euler paths and circuits, we’re primarily interested in whether an Euler path or circuit exists .
Why do we care if an Euler circuit exists? Think back to our housing development lawn inspector from the beginning of the chapter. The lawn inspector is interested in walking as little as possible. The ideal situation would be a circuit that covers every street with no repeats. That’s an Euler circuit! Luckily, Euler solved the question of whether or not an Euler path or circuit will exist.
A graph will contain an Euler path if it contains at most two vertices of odd degree.
A graph will contain an Euler circuit if all vertices have even degree
In the graph below, vertices A and C have degree 4, since there are 4 edges leading into each vertex. B is degree 2, D is degree 3, and E is degree 1. This graph contains two vertices with odd degree (D and E) and three vertices with even degree (A, B, and C), so Euler’s theorems tell us this graph has an Euler path, but not an Euler circuit.
Now we know how to determine if a graph has an Euler circuit, but if it does, how do we find one? While it usually is possible to find an Euler circuit just by pulling out your pencil and trying to find one, the more formal method is Fleury’s algorithm.
- Start at any vertex if finding an Euler circuit. If finding an Euler path, start at one of the two vertices with odd degree.
- Choose any edge leaving your current vertex, provided deleting that edge will not separate the graph into two disconnected sets of edges.
- Add that edge to your circuit, and delete it from the graph.
- Continue until you’re done.
Let’s find an Euler Circuit on this graph using Fleury’s algorithm, starting at vertex A.
Solution
Step 1:
Original Graph. Choosing edge AD.
Circuit so far: AD
Step 2:
AD deleted. D is current. Can’t choose DC since that would disconnect graph. Choosing DE
Circuit so far: ADE
Step 3:
E is current. From here, there is only one option, so the rest of the circuit is determined.
Circuit: ADEBDCA
Does the graph below have an Euler Circuit? If so, find one.
- Answer
-
Yes, all vertices have even degree so this graph has an Euler Circuit. There are several possibilities. One is: ABEGFCDFEDBCA
Not every graph has an Euler path or circuit, yet our lawn inspector still needs to do her inspections. Her goal is to minimize the amount of walking she has to do. In order to do that, she will have to duplicate some edges in the graph until an Euler circuit exists.
Eulerization is the process of adding edges to a graph to create an Euler circuit on a graph. To eulerize a graph, edges are duplicated to connect pairs of vertices with odd degree. Connecting two odd degree vertices increases the degree of each, giving them both even degree. When two odd degree vertices are not directly connected, we can duplicate all edges in a path connecting the two.
Note that we can only duplicate edges, not create edges where there wasn’t one before. Duplicating edges would mean walking or driving down a road twice, while creating an edge where there wasn’t one before is akin to installing a new road!
For the rectangular graph shown, three possible eulerizations are shown. Notice in each of these cases the vertices that started with odd degrees have even degrees after eulerization, allowing for an Euler circuit.
In the example above, you’ll notice that the last eulerization required duplicating seven edges, while the first two only required duplicating five edges. If we were eulerizing the graph to find a walking path, we would want the eulerization with minimal duplications.
Eulerize the graph shown, then find an Euler circuit on the eulerized graph.
- Answer
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This graph can be eulerized by duplicating the edge BC, as shown. One possible Euler circuit on the eulerized graph is ACDBCBA
Looking again at the graph for our lawn inspector, the vertices with odd degree are shown highlighted. With eight vertices, we will always have to duplicate at least four edges.
Solution
In this case, we need to duplicate five edges since two odd degree vertices are not directly connected. Without weights we can’t be certain this is the eulerization that minimizes walking distance, but it looks pretty good.
The problem of finding the optimal eulerization is called the Chinese Postman Problem, a name given by an American in honor of the Chinese mathematician Mei-Ko Kwan who first studied the problem in 1962 while trying to find optimal delivery routes for postal carriers. This problem is important in determining efficient routes for garbage trucks, school buses, parking meter checkers, street sweepers, and more.
Unfortunately, algorithms to solve this problem are fairly complex.
[1] Bogdan Giuşcă. http://en.Wikipedia.org/wiki/File:Ko...rg_bridges.png
[2] Sam Beebe. http://www.flickr.com/photos/sbeebe/2850476641/ , CC-BY