   Chapter 10.1, Problem 45ES ### 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 Lemma 10.1.1(b): If vertices v and w are part of a circuit in a graph G and one edge is removed from the circuit, then there still exists a trail from v to w in G.

To determine

If vertices v and w are part of a circuit in a graph G and one edge is removed from the circuit, then there still exists a trail from v to w in G.

Explanation

Given information:

vertices v and w are part of a circuit in a graph G and one edge is removed from the circuit.

Proof:

Let v and w be vertices in a circuit ve1v1e2v2....vn1enwen+1vn+1en+2vn+2...vn+m1en+mv of the graph G (for some positive integer n and some positive integer m ) and let us remove the edge ei from the circuit (1im+n).

It is safe to assume that v and w are distinct, because if v = w, then either ve1v1e2v2...vn1enw or ven+mvn+m1en+m2vn+m2...vn+1en+1w will still from a trial from v to w (as the circuit contain no repeated edges)

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