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Chapter 10: Problem 28

How do you determine whether or not a graph is traversable?

Short Answer

Expert verified
A graph is traversable if all vertices have an even degree (Eulerian graph), or if there are exactly two vertices with odd degrees (semi-Eulerian); start at one odd-degree vertex and end at the other. In cases where more than two vertices have odd degrees, the graph is not traversable.

Step by step solution

01

Understand traversability

A graph is called traversable if it is possible to navigate through the graph, visiting each edge exactly once. This is linked to the property of Eulerian graphs (named after Leonard Euler), which are graphs where each vertex has an even degree.
02

Calculate the degree of vertices

To determine whether a graph is traversable you should determine the degree of each vertex. The degree of a vertex is the number of edges that are incident to (or meet at) that vertex. This can be done by examining the graph and counting the number of edges that each vertex has.
03

Determine traversability

A graph is traversable if all vertices in the graph have an even degree (Eulerian graph). However, if exactly two vertices have odd degrees and the rest have even degrees, the graph is semi-Eulerian, meaning it is traversable but one must start at one of the odd-degree vertices and end at the other. If more than two vertices have odd degrees, the graph is not traversable.

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