   Chapter 10.6, Problem 22ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove that an edge e is contained in every spanning tree for a connected graph G if, and only if, removal of e disconnects G.

To determine

To prove:

There is an edge e which is contained in each spanning tree of a connected graph G if, and only if, removal of this edge e disconnects the connected graph G.

Explanation

Given information:

An edge e is contained in every spanning tree for a connected graph G.

Concept used:

Sub graph of G obtained by removing e is G'.

Proof:

Let G be a connected graph.

Suppose e is an edge contained in every spanning tree for G.

If possible, suppose that the graph obtained by removing e from G is connected.

Let G' be the sub graph of G obtained by removing e. Then G' is connected.

Let T' be the spanning tree of G'.

Then T' is also a spanning tree of G because G' contains all vertices of G except the edge e

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