   # Transportation For the train routes given in Exercise 1, is it possible to travel along all of the train routes without traveling along any route twice? Explain how you reached your conclusion. ### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584

#### Solutions

Chapter
Section ### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 5.1, Problem 25ES
Textbook Problem
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## Transportation For the train routes given in Exercise 1, is it possible to travel along all of the train routes without traveling along any route twice? Explain how you reached your conclusion.

To determine

To determine if it is possible to travel along all of the train routes without traveling along any route twice.

### Explanation of Solution

Given information:

Given, an "X" in the given table indicates a direct train route between the corresponding cities.

A graph is to be made where each vertex represents a city and an edge connects two vertices if there is a train route between the corresponding cities.

And the graph is,

Calculation:

It is possible to travel along all of the train routes without traveling along any route twice if the graph satisfies the condition of either Euler circuit or Euler path.

An Eulerian graph (that starts and ends at the same vertex) is said to be Euler circuit if it uses every edge, but only once. (i.e. no edge is used more than once).

Euler path is defined as the path in a connected graph that has exactly two vertices of odd degree with all the remaining vertices of even degree and the path starts at one of the vertices with odd degree and ends at the other vertex with odd degree.

Firstly, the graph is checked to know whether it satisfies to be an Eulerian graph.

A graph is said to be Eulerian graph if it is a connected graph and every vertex of the graph has even degree...

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