1 / 24

Section 14.2 Euler Paths, and Euler Circuits

Section 14.2 Euler Paths, and Euler Circuits. What You Will Learn. Euler Paths Euler Circuits Euler’s Theorem Fleury’s Algorithm. Euler Path. An Euler path is a path that passes through each edge of a graph exactly one time. Euler Circuit.

ebner
Download Presentation

Section 14.2 Euler Paths, and Euler Circuits

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 14.2Euler Paths, and Euler Circuits

  2. What You Will Learn • Euler Paths • Euler Circuits • Euler’s Theorem • Fleury’s Algorithm

  3. Euler Path • An Euler path is a path that passes through each edge of a graph exactly one time.

  4. Euler Circuit • An Euler circuit is a circuit that passes through each edge of a graph exactly one time.

  5. Euler Path versus Euler Circuit • The difference between an Euler path and an Euler circuit is that an Euler circuit must start and end at the same vertex.

  6. Euler Path versus Euler Circuit • Euler Path D, E, B, C, A, B, D, C, E Euler Circuit D, E, B, C, A, B, D, C, E, F, D

  7. Euler’s Theorem • For a connected graph, the following statements are true: 1. A graph with no odd vertices (all even vertices) has at least one Euler path, which is also an Euler circuit. An Euler circuit can be started at any vertex and it will end at the same vertex.

  8. Euler’s Theorem 2. A graph with exactly two odd vertices has at least one Euler path but no Euler circuits. Each Euler path must begin at one of the two odd vertices, and it will end at the other odd vertex. 3. A graph with more than two odd vertices has neither an Euler path nor an Euler circuit.

  9. Example 3: Solving the Königsberg Bridge Problem • Could a walk be taken through Königsberg • during which • each bridge is • crossed exactly • one time?

  10. Example 3: Solving the Königsberg Bridge Problem • Solution • Here’s a representation of the problem: vertices are the land, edges are the bridges.

  11. Example 3: Solving the Königsberg Bridge Problem • Solution • Does an Euler path exist? • Four odd vertices: A, B, C, D • So, according to item 3 ofEuler’s Theorem, no Eulerpath exists.

  12. Fleury’s Algorithm • To determine an Euler path or an Euler circuit 1. Use Euler’s theorem to determine whether an Euler path or an Euler circuit exists. If one exists, proceed with steps 2-5.

  13. Fleury’s Algorithm 2. If the graph has no odd vertices (therefore has an Euler circuit, which is also an Euler path), choose any vertex as the starting point. If the graph has exactly two odd vertices (therefore has only an Euler path), choose one of the two odd vertices as the starting point.

  14. Fleury’s Algorithm 3. Begin to trace edges as you move through the graph. Number the edges as you trace them. Since you can’t trace any edges twice in Euler paths and Euler circuits, once an edge is traced consider it “invisible.”

  15. Fleury’s Algorithm 4. When faced with a choice of edges to trace, if possible, choose an edge that is not a bridge (i.e., don’t create a disconnected graph with your choice of edges). 5. Continue until each edge of the entire graph has been traced once.

  16. Example 6: Crime Stoppers Problem • On the next slide is a representation of the Country Oaks subdivision of homes. The Country Oaks Neighborhood Association is planning to organize a crime stopper group in which residents take turns walking through the neighborhood with cell phones to report any suspicious activity to the police.

  17. Example 6: Crime Stoppers Problem

  18. Example 6: Crime Stoppers Problem • a) Can the residents of Country Oaks start at one intersection (or vertex) and walk each street block (or edge) in the neighborhood exactly once and return to the intersection where they started? • b) If yes, determine a circuit that could be followed to accomplish their walk.

  19. Example 6: Crime Stoppers Problem • Solution • a) Does a Euler circuit exist? • There are no odd vertices. By item 1 ofEuler’s theorem,there is at leastone Euler circuit.

  20. Example 6: Crime Stoppers Problem • Solution • Start at A. Choose AB or AE, neither is a bridge. Choose AB.

  21. Example 6: Crime Stoppers Problem • Solution • Continue to trace from vertex to vertex around the outside of the graph. Notice that no edge chosen is a bridge.

  22. Example 6: Crime Stoppers Problem • Solution • At vertex E. EA is a bridge, so choose either EB or EI. Choose EB. • Now from vertex B we must choose BF.

  23. Example 6: Crime Stoppers Problem • Solution • From vertex F, FI is a bridge, so we must choose either FC or FI. Choose FC. • From vertex C, our only choice is CG.

  24. Example 6: Crime Stoppers Problem • Solution • From vertex G, GJ is a bridge, so we must choose GG. • Back at vertex G, and from now on, there is only one choice at each vertex.

More Related