Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 14 - Graph Theory - Chapter 14 Test: 10

Answer

(a) We can use a graph to model the layout of the city. (b) Since the graph has exactly two odd vertices, the graph has at least one Euler path. By following an Euler path, it is possible to cross each bridge exactly once. (c) A route which crosses each bridge exactly once should start at either Island A or Island B.
1514515892

Work Step by Step

(a) We can use a graph to model the layout of the city. Each vertex represents one land mass and each edge represents a bridge between two land masses. (b) We need to verify the number of odd vertices in the graph. Vertex Island A and vertex Island B are odd vertices. The other vertices are even. Since the graph has exactly two odd vertices, the graph has at least one Euler path. An Euler path is a path which travels through every edge of the graph exactly once. Each edge in the graph represents a bridge. By following an Euler path, it would be possible to cross each bridge exactly once. Since this graph has at least one Euler path, it is possible to find a path that crosses each bridge exactly once. (c) When a graph has exactly two odd vertices, any Euler path starts at one odd vertex and ends at the other odd vertex. Therefore, a route which crosses each bridge exactly once should start at either Island A or Island B.
Small 1514515892
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.