Question

Prove that the following problem is NP-complete: given an undirected graph G = (V, E) and an integer k, return a clique of size k as well as an independent set of size k, provided both exist.

Answer 1

The answer to the present question is yes if G has both of those subsets.As we all know that both the issues i.e. clique and independent set are NP-Complete. Also, we all know that the verification of this problem, is in NP.

Performing a discount on the above problem is that the problem (which contains both independent sets and cliques) to either a drag consisting entirely of cliques or independent sets. Since we don't skills this reduction are often performed without the lose of any information which will be needed to scale back the reduction back to its original form. Or we will also say that,

Since we all know that the clique and independent-set problems are NP-Complete, we will use that as a base for proving the matter . Let's call our problem CC.

Assume we've been given a graph G which features a clique C of size k. Reduce the graph into a graph G' (read: G prime) which features a clique C' of size k' and independent-set I of size k' by attaching k vertices to every vertex in C. The reduction is occuring in polynomial time because the vertices addition takes O(n*k) time (n vertices within the graph and k vertices attached to every node).(Note that C=C' and k=k'.)Now Assume a graph G' which features a clique C' of size k' and independent-set I of size k' which is decided to be true. The reduction to the clique problem is trivial because we don't need to improvise the graph to seek out only a clique.

An easier way of reduction from Clique to CC is to require your input graph for Clique and add the k isolated vertices thereto . (If k is present within the input in binary, adding k extra vertices is an exponential amount of labor relative to the length of the input. So first make sure k is at the most the amount of vertices within the graph: if k is bigger , produce any small "no" instance, e.g., the graph on one vertex.)