Thinking Mathematically (6th Edition)

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

Chapter 14 - Graph Theory - 14.4 Trees - Exercise Set 14.4: 56

Answer

The edges AB, CD, EF, BF, and CF make a spanning tree. The edges AB, CD, EF, BF, and BC make a spanning tree. The edges AB, CD, EF, BC, and CF make a spanning tree.

Work Step by Step

A spanning tree includes all the vertices in the original graph, is connected, and does not have any circuits. If the original graph has $n$ vertices, then the spanning tree has $n$ vertices and $n-1$ edges. In this exercise, the original graph has 6 vertices so a spanning tree must have 6 vertices and 5 edges. Clearly, the edges AB, CD, and EF must be included in the spanning tree, otherwise the spanning tree will not be connected. We then need to include 2 of the remaining edges to complete the spanning tree. The edges AB, CD, EF, BF, and CF make a spanning tree. The edges AB, CD, EF, BF, and BC make a spanning tree. The edges AB, CD, EF, BC, and CF make a spanning tree. There are three spanning trees for the given graph in this exercise.
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