Unformatted text preview: Instance: An undirected graph G = ( V,E ) and an integer 0 ≤ k ≤  V  . Question: Is there a subset I ⊆ V of vertices such that  I  ≥ k and no two vertices in I are adjacent ( independent set )? [Hint: If C is a vertex cover for G , what can you say about VC ? If I is an independent set for G , what can you say about VI ?] (not in text) Give a polynomial reduction from INDEPENDENT SET to CLIQUE. [Hint: The complement of an undirected graph G = ( V,E ) is the graph G = ( V,E ) where ( u,v ) ∈ E iﬀ ( u,v ) / ∈ E , for all distinct vertices u,v ∈ V .] (not in text) (more challenging) Give a polynomial reduction from (directed) HC to HAMCYCLE (the undirected version of the Hamiltonian path problem (page 979)). [Hint: Given an instance G of HC, build a new graph G by replacing each vertex of G with a path of length 2. Connect endpoints of diﬀerent paths together with edges in some way based on the edges of G .] 1...
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 Fall '11
 Fenner
 Algorithms, Graph Theory, Glossary of graph theory, independent set, Graph theory objects, polynomial reduction

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