Given a graph with n vertices, deciding if there exists a clique of size ≥ 195 is : 

The correct answer is NP-Complete.

Key Points

• The problem of deciding whether a given graph contains a clique of size ≥ k is a well-known problem in computer science.
• This problem is a generalization of the clique problem, which is one of Karp's 21 NP-complete problems.
• A clique in an undirected graph is a subset of vertices such that every two distinct vertices in the clique are adjacent.
• The decision version of the clique problem, which asks if there is a clique of at least a certain size, is NP-complete.
• Since the problem of finding a clique of size ≥ 195 is a specific case of this general problem, it is also NP-complete.

Additional Information

• NP-complete problems are those that are both in NP and NP-hard. A problem is in NP if a given solution can be verified in polynomial time.
• A problem is NP-hard if every problem in NP can be reduced to it in polynomial time.
• Finding a clique of size ≥ 195 is challenging because the number of potential cliques grows exponentially with the size of the graph.
• Many approximation algorithms and heuristics exist to tackle NP-complete problems, but no polynomial-time solution is known for solving them exactly.
• Understanding the complexity class of a problem helps in determining the computational resources needed to solve it and guides the choice of algorithms.