## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 14 - Graph Theory - 14.4 Trees - Exercise Set 14.4: 19

#### Answer

We can find a spanning tree by removing edge AC and edge DE from the original graph. The modified graph will have 6 vertices and 5 edges. Also, the modified graph will still be connected and there will be no circuits. Therefore, the modified graph will be a tree.

#### Work Step by Step

One characteristic of a tree is the following: If the graph has $n$ vertices, then the graph has $n-1$ edges. The original graph has 6 vertices and 7 edges. Therefore this graph is not a tree. To make a spanning tree, we need to remove 2 edges from the original graph. By removing edge AC and edge DE, the graph will have 6 vertices and 5 edges. Also, the graph will still be connected and there will be no circuits. Therefore, the modified graph will be a tree. This is one spanning tree, but other spanning trees are possible.

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