The Reduction Argument
We must show
- F satisfiable implies G has a clique of size k.
- Given a satisfying assignment for F, for each clause pick a literal that is satisfied. Those literals in the graph G form a k-clique.
- G has a clique of size k implies F is satisfiable.
- Given a k-clique in G, assign each literal in the clique to be 1. This yields a satisfying assignment to F.