To Prove: - Clique is an NPC or not?
For this you have to satisfy the following below-mentioned points: -
Definition: - In Clique, every vertex is directly connected to another vertex, and the number of vertices in the Clique represents the Size of Clique.
CLIQUE COVER: - Given a graph G and an integer k, can we find k subsets of verticesV1, V2...VK, such that UiVi = V, and that each Vi is a clique of G.
The following figure shows a graph that has a clique cover of size 3.
2)3CNF ≤ρ Clique
Proof:-For the successful conversion from 3CNF to Clique, you have to follow the two steps:-
Draw the clause in the form of vertices, and each vertex represents the literals of the clauses.
Clique ≤ρ 3CNF
Proof: - As you know that a function of K clause, there must exist a Clique of size k. It means that P variables which are from the different clauses can assign the same value (say it is 1). By using these values of all the variables of the CLIQUES, you can make the value of each clause in the function is equal to 1
Example: - You have a Boolean function in 3CNF:-
(X+Y+Z) (X+Y+Z') (X+Y'+Z)
After Reduction/Conversion from 3CNF to CLIQUE, you will get P variables such as: - x +y=1, x +z=1 and x=1
Put the value of P variables in equation (i)
(1)(1)(1)=1 output verified
4) Clique ϵ NP:-
Proof: - As you know very well, you can get the Clique through 3CNF and to convert the decision-based NP problem into 3CNF you have to first convert into SAT and SAT comes from NP.
So, concluded that CLIQUE belongs to NP.
Proof of NPC:-