Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. k For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. a In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? b In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.

Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. k For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. a In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? b In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is
incident to at least one vertex in C. We say that a set of vertices I form an independent set if
no edge in G connects two vertices from I.
For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of
the 20 edges in the graph has at least one endpoint in C, and I = = [a, c, i, k] is an
independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik.
2a
In the example above, notice that each vertex belongs to the vertex cover C or the independent
set I. Do you think that this is a coincidence?
2b
In the above graph, clearly explain why the maximum size of an independent set is 5. In other
words, carefully explain why there does not exist an independent set with 6 or more vertices.

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