Q. For a given graph G = (V,E), a subset SCVIS called an independent set if (u,v) E E for every pair of vertices u, v E S. The three independent set problems ISD, ISO and ISF all have as input a graph G=(V,E). ISD also has as part of the input a natural number k. ISD is a decision problem: the question is whether or not there is a subset S of V with at least k elements such that (u,v)£E for every pair of vertices u, v ES? ISO and ISF do not have yes/no answers. The answer to ISO is an integer n– size of the largest independent set in graph G. The answer to ISF is subset S – largest independent set in graph G. Show that all three of these problems are polynomial-time Turing equivalent. I.e. show the solution of ISO and ISF that uses ISD as a subroutine. Also show a solution to ISD that uses ISO as a subroutine. Provide detailed pseudocode solutions with justification of correctness and running time.

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