Launch problem: Lining streets

Transcription

1 Math 5340 June 15,2012 Dr. Cordero Launch problem: Lining streets Lining Street Problem A Problem on Eulerian Circuits Your job for the day is to drive slowly around the neighborhood. Why? You'll be driving the truck that paints lines on streets. The truck is set up to paint only center lines. You'll be painting all streets in the residential neighborhood next to the street maintenance barn at which you'll start and finish your work. To the right is a map of that neighborhood. The question is What route should you follow to avoid more driving than necessary? 1. What mathematical objects might represent the situation of this problem? You could use a graph with vertices representing corners and edges representing streets between them, perhaps something like this: The question now would be: What's a sequence of edges that includes all edges in this graph once and only once and begins and ends at the vertex B? If a trail includes every edge of the graph, it's called an Eulerian trail, after the mathematician Euler (pronounced "OY-ler"). An Eulerian trail that begins and ends at the same vertex is an Eulerian circuit. Explore: Expansion Suppose a new street is built in the neighborhood, and it's represented on the graph by an edge between B and F. (It would intersect the road from A to E at a new vertex.) 3. Explain how the additional edge affects the existence of an Eulerian circuit.

2 Summary: Comparing solutions Here are three proposed Eulerian circuits for the graph : BEDCBAHEFGAB BAGFAEDCB BEHAFHBCDEBAGFHB 4. Critique each of these proposals Abstractions: Brainstorming One problem raised in Summary was Under what conditions does a graph contain an Eulerian circuit? an Eulerian trail? A graph that contains an Eulerian circuit is called an Eulerian graph. 5. Brainstorm about ideas and questions concerning properties of graphs that contain Eulerian circuits and Eulerian trails. Abstractions: Degree conditions Someone wrote: Here's a conjecture and, I think, a proof. The conjecture is that, if a graph has an Eulerian circuit, then each vertex has even degree. My proof: For each vertex, the entries to that vertex can be paired up with the exits. So each vertex must touch an even number of edges. 6. (Homework) Critique the proposed conjecture and proof. Do you agree with the conjecture? Is the proof adequate? Compare with Theorem Abstractions: Non-Eulerian graphs What can you say about graphs that are not Eulerian? We know if a graph is Eulerian, then all its vertices have even degree. Can we conclude that if a graph is not Eulerian, then at least one vertex must have odd degree? 7. Do you think that if a graph is not Eulerian, then at least one vertex must have odd degree? Why or why not? 8. What conditions on a graph would allow an Eulerian trail between two vertices but not an Eulerian circuit? Abstractions: A proof? One person wrote:

3 I conjectured that, if every vertex of a graph has even degree, then the graph is Eulerian. And I think I have a proof of that conjecture. Begin walking at any vertex along any incident edge. Keep going as long as you can without repeating edges. Every vertex has even degree, so if you get there you will be able to leave. Except you might get back to the first vertex and find all edges used up. If you do that, and you've missed some edges, then on the path you've traveled there has to be some vertex incident to an untraveled edge. In fact, because the degree of that vertex is even, there must be at least two untraveled edges touching that vertex. Start at that vertex and walk along untraveled edges. By the same reasoning as before, you can leave any vertex you get to except perhaps the one you began at. Rats! Which one? Well, I'd better give it a name. Say the vertex you began the whole thing at is called v. You make your first circuit. Now suppose there's a vertex on that circuit with unused edges. Call that vertex w. You make a new circuit starting and ending at w. Now you can put the circuits together into a new circuit that starts at v and goes to w, then goes around the second circuit until it's back to w,, then goes home to v. If there are still edges you haven't walked around, you can find another vertex with unused edges on this newest circuit, so splice in a third circuit starting and stopping there. You can keep going until you get a circuit that includes all edges. I think. (Questions 9-12 below are Homework.) 9. How does this conjecture relate to Theorem ? 10. Rewrite the argument in a clearer way. Applications: Robot tracks Here we examine several problems that help answer the question posed in Summary: What other problems can be modeled using Eulerian circuits? You are an engineer for a small robotics company. The robots that your company makes are like small railroad cars. They travel along special tracks and are designed to pick up and deliver bins in which a large variety of items can be placed. As part of your regular maintenance, you visit a client to inspect the robot tracks your company Installed last year. The tracks follow this plan:

4 You must walk along the tracks to inspect them, and you don't want to walk along any track more than once if you can avoid doing so. 11. Can you inspect all of the tracks by walking along each one once and only once, ending up at your starting point? If so, how? If not, why not? Applications: A proposed solution One person wrote You can obviously make an Eulerian circuit on the graph that represents the situation. In fact, you could add two more tracks and still have a graph with an Eulerian circuit. 12. Critique this idea. How "obvious" do you think it is that the graph contains an Eulerian circuit? 13. Do you agree with the claim about two more tracks. Explain. Applications: Trick or treat Here's a different problem: You are planning to drive some kids through your neighborhood as they trick-or-treat on Halloween. Actually, you'll be driving on the streets as they walk door-to-door and back and forth across the streets. You agree to drive by all houses in the neighborhood represented by the map to the right, and you want to avoid backtracking if you can. 14. How could you plan the route to minimize the amount of driving? Applications: One idea

5 One person proposed this trail: EFGHMLDEABCGKJIE 15. Critique this proposal. Applications: Bridges of Königsberg Most mathematical historians trace the origins of graph theory to a problem posed by Leonhard Euler (pronounced "OY-ler") in Here is Euler's description and picture of the problem: The problem is stated as follows: In the town of Königsberg in Prussia there is an island A, called "Kneiphof," with the two branches of the river (Pregel) flowing around it, as shown in the figure. There are seven bridges crossing the two branches. The question is whether a person can plan a walk in such a way that he will cross each of these bridges once, but not more than once. I was told that while some denied the possibility of doing this and others were in doubt, there were none who maintained that it was actually possible. 16. How can the Bridges of Königsberg problem be modeled and solved using graph theory? In particular, what would the edges and vertices represent? Application: Discussion of Königsberg As with the earlier puzzle, there's one major difficulty in modeling the Bridges of Königsberg: if each region is represented by a single vertex, then there must be multiple edges (representing bridges) between two vertices. So the model would be a pseudograph, not a graph. You can get around the difficulty as follows: As with the puzzle, you can insert some additional vertices to have a simple graph. 17. Will inserting a vertex on an existing edge ever affect whether or not the graph is Eulerian?