   Chapter 10.1, Problem 50ES ### 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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# Let G be a connected graph, and let C be any circuit in G that does not contain every vertex of C. Let G’ be the subgraph obtained by removing all the edges of C from G and also any vertices that become isolated when the edges of C are removed. Prove that there exist a vertex v such that v is in both C and G’.

To determine

To prove:

Prove that there exists a vertex v such that v is in both C and G.

Explanation

Given information:

Let G be a connected graph, and let C be any circuit in G that does not contain every vertex of C. Let G’ be the subgraph obtained by removing all the edges of C from G and also any vertices that become isolated when the edges of C are removed.

Calculation:

Given:

G is a connected graph

C is a circuit in G that does not contain every vertex of G

G is the subgraph obtained by removing all edges of C from G and removing all isolated vertices (due to the removal of edges)

Since C is a circuit in G that does not contain every vertex in G, there exists a vertex w in G that is not contained in the circuit C.

Since the graph G is connected, there exists an edge e (that is not a loop) with w as its endpoint and this edge e cannot be a part of the circuit C (if e is in the circuit C, then w would need to be in the circuit C as well). Note that this implies that w is a vertex in G and e is also an edge in G

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