[Solved]6 Independent Set Problem Given Graph G Integer K Set S K Vertices G Two Vertices S Adjace Q37220098

6. The Independent Set Problem is: Given a graph G and an integer K, is there a set S of K vertices in G such that no two vertices in S are adjacent? The Exactly Half Independent Set Problem is: Given a graph G on n vertices where n is even, is there a set S of n/2 vertices in G such that no two vertices in S are adjacent? (a) Prove that the Exactly Half Independent Set Proble is in NP (b) Prove that the Independent Set Proble reduces to the Exactly Half Independent Set Problem in polynomial time. (Hint: Given a graph G in the Independent Set Problem, add vertices and edges to turn the problem into the Exactly Half Independent Set Problem. For your reduction, you will need to consider separately the cases of independent problems where n >2K from the cases where ns 2k.) Note that proving a) and b) proves that the Exactly Half Independent Set Problem is NP-complete.) Show transcribed image text 6. The Independent Set Problem is: Given a graph G and an integer K, is there a set S of K vertices in G such that no two vertices in S are adjacent? The Exactly Half Independent Set Problem is: Given a graph G on n vertices where n is even, is there a set S of n/2 vertices in G such that no two vertices in S are adjacent? (a) Prove that the Exactly Half Independent Set Proble is in NP (b) Prove that the Independent Set Proble reduces to the Exactly Half Independent Set Problem in polynomial time. (Hint: Given a graph G in the Independent Set Problem, add vertices and edges to turn the problem into the Exactly Half Independent Set Problem. For your reduction, you will need to consider separately the cases of independent problems where n >2K from the cases where ns 2k.) Note that proving a) and b) proves that the Exactly Half Independent Set Problem is NP-complete.)

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Answer to 6. The Independent Set Problem is: Given a graph G and an integer K, is there a set S of K vertices in G such that no tw… . . .