The.Algorithm.Design.Manual.Springer-Verlag.1998

Clique and Independent Set
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Clique and Independent Set
Consider the clique problem, further discussed in Section :
Figure: A small graph with a five-vertex clique
Input: A graph G=(V,E) and integer .
Output: Does the graph contain a clique of j vertices; i.e. is there a subset , where , such that every pair
of vertices in S defines an edge of G? For example, the graph in Figure contains a clique of five vertices. In the
independent set problem, we looked for a subset S with no edges between two vertices of S. However, for a clique,
we insist that there always be an edge between two vertices. A reduction between these problems results by
reversing the roles of edges and non-edges, an operation known as complementing the graph:
IndependentSet(G,k)
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Construct a graph G=(V',E') where V'=V, and
Return the answer to Clique(G',k)
For all (i,j) not in E, add (i,j) to E'