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3. Construct an example of a graph on n vertices such that running this algorithm on the vertices in one order yields an independent set of size 1, and processing the vertices in a different order yields an independent set of size n – 1, which is maximal.

3. Construct an example of a graph on n vertices such that running this algorithm on the vertices in one order yields an independent set of size 1, and processing the vertices in a different order yields an independent set of size n – 1, which is maximal.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 80EQ
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H. I only need question 3 done! It involves discrete math!

Let's consider an undirected graph G that for any vertices, i, j E I and there is no edge between i and į in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent set is not necessarily the largest independent set in G. Let a(G) denote the size of the largest maximal independent set in G. = (V,E). An independent subset is a subset I cV such
Transcribed Image Text:Let's consider an undirected graph G that for any vertices, i, j E I and there is no edge between i and į in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent set is not necessarily the largest independent set in G. Let a(G) denote the size of the largest maximal independent set in G. = (V,E). An independent subset is a subset I cV such
Let's consider the following algorithm for generating maximal independent sets: starting with an empty set I, process the vertices in V one at a time, adding v to I is v not connected to any vertex already in I. 2. Argue that the output I of this algorithm is a maximal independent set 3. Construct an example of a graph on n vertices such that running this algorithm on the vertices in one order yields an independent set of size 1, and processing the vertices in a different order yields an independent set of size n – 1, which is maximal.
Transcribed Image Text:Let's consider the following algorithm for generating maximal independent sets: starting with an empty set I, process the vertices in V one at a time, adding v to I is v not connected to any vertex already in I. 2. Argue that the output I of this algorithm is a maximal independent set 3. Construct an example of a graph on n vertices such that running this algorithm on the vertices in one order yields an independent set of size 1, and processing the vertices in a different order yields an independent set of size n – 1, which is maximal.
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