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In this question you will prove that the complete graph with n vertices Kn is the only graph on n vertices with vertex connectivity equal to n - 1. Let G be a graph with n vertices. 1. Prove that if removing n - 2 vertices from G disconnects G then the vertex connectivity of G is at most n - 2. 2. Prove that if G is not equal to Kn then the vertex connectivity of G is at most n - 2. Hint: if G# Kn then there are two vertices of G that do not have an edge between them. Note that, since Kn has vertex connectivity equal to n - 1 (you do not have to prove this), this means that it is the only graph on n vertices with vertex connectivity equal to n - 1.

In this question you will prove that the complete graph with n vertices Kn is the only graph on n vertices with vertex connectivity equal to n - 1. Let G be a graph with n vertices. 1. Prove that if removing n - 2 vertices from G disconnects G then the vertex connectivity of G is at most n - 2. 2. Prove that if G is not equal to Kn then the vertex connectivity of G is at most n - 2. Hint: if G# Kn then there are two vertices of G that do not have an edge between them. Note that, since Kn has vertex connectivity equal to n - 1 (you do not have to prove this), this means that it is the only graph on n vertices with vertex connectivity equal to n - 1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In this question you will prove that the complete graph with n vertices Kn is the only graph on n vertices with vertex connectivity equal to n - 1. Let G be a graph with n vertices. 1. Prove that if removing n - 2 vertices from G disconnects G then the vertex connectivity of G is at most n - 2. 2. Prove that if G is not equal to Kn then the vertex connectivity of G is at most n — 2. Hint: if G# Kn then there are two vertices of G that do not have an edge between them. Note that, since Kn has vertex connectivity equal to n - - 1 (you do not have to prove this), this means that it is the only graph on n vertices with vertex connectivity equal to n - 1.
Transcribed Image Text:In this question you will prove that the complete graph with n vertices Kn is the only graph on n vertices with vertex connectivity equal to n - 1. Let G be a graph with n vertices. 1. Prove that if removing n - 2 vertices from G disconnects G then the vertex connectivity of G is at most n - 2. 2. Prove that if G is not equal to Kn then the vertex connectivity of G is at most n — 2. Hint: if G# Kn then there are two vertices of G that do not have an edge between them. Note that, since Kn has vertex connectivity equal to n - - 1 (you do not have to prove this), this means that it is the only graph on n vertices with vertex connectivity equal to n - 1.
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