Thinking Mathematically (6th Edition)

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

Chapter 14 - Graph Theory - 14.2 Euler Paths and Euler Circuits - Exercise Set 14.2: 73

Answer

We should build an additional bridge to connect the two islands. This additional bridge connecting the two islands will allow us to walk through the city while crossing each bridge exactly once.

Work Step by Step

In the original graph, all four vertices are odd vertices. Therefore, the graph does not have an Euler path. If we add one edge to the graph connecting any two vertices, then both of these vertices will be even vertices. This modified graph will have two odd vertices and two even vertices. Since the modified graph will have exactly two odd vertices, the graph will have at least one Euler path. An Euler path is a path that travels through every edge on the graph exactly once. If we follow an Euler path across the bridges in the city, then we will be able to cross every bridge in the city exactly once. In a graph with exactly two odd vertices, any Euler path starts at one odd vertex and end at the other odd vertex. To walk through the city, we need to begin at one bank and end at the other bank. Therefore, we should build an additional bridge to connect the two islands. If we build the bridge between the two islands, then the two islands will be even vertices on the graph, and the two banks will still be odd vertices on the graph. Then any Euler path starts at one bank and ends at the other bank. This additional bridge connecting the two islands will allow us to walk through the city while crossing each bridge exactly once.
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