   Chapter 10.6, Problem 24ES ### 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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# Suppose that T is a minimum spanning tree for a connected, weighted graph G and that G contains an edge e (not a loop) that is not in T. Let v and w be the endpoints of e. By exercise 18 there is a unique path in T from v to w. Let e’ be any edge of this path. Prove that w ( e ′ ) ≤ w ( e ) .

To determine

To prove:

w(e')w(e).

Explanation

Given information:

Suppose, there is a minimum spanning tree T for a connected, weighted graph G.

The connected graph contains an edge e which is not in spanning tree T.

Concept used:

Correctness of Prim’s Algorithm:

When a connected, weighted graph G is input to prim’s algorithm, the output is a minimum spanning tree for G.

Proof:

Consider the end points of the edge are v and w.

The objective is to prove that w(e')w(e).

Assume that w(e')>w(e).

Form a new graph T' by adding e to T and deleting e'.

The addition of an edge to a spanning tree creates a circuit, and the deletion of an edge from a circuit does not disconnect a graph.

Consequently T' is also a spanning tree for G

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