Is the clique problem NP-complete?

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Is the clique problem NP-complete?

In this Wikipedia article about the Clique problem in graph theory it states in the beginning that the problem of finding a clique of size K, in a graph G is NP-complete:
What is the clique decision problem?
Correspondingly, the Clique Decision Problem is to find if a clique of size k exists in the given graph or not. To prove that a problem is NP-Complete, we have to show that it belongs to both NP and NP-Hard Classes.

What is the correct abbreviation for the clique problem?

Lets use abbreviation CP for the Clique problem. IS is the Independent Set problem. It suffices to show (a) the clique problem is in NP, and (b) IS ≤ p CP. Part (a) is easy.
Is it possible to solve the maximum clique problem?
Because of the hardness of the decision problem, the problem of finding a maximum clique is also NP-hard. If one could solve it, one could also solve the decision problem, by comparing the size of the maximum clique to the size parameter given as input in the decision problem.

What is the clique problem in CS?

But in this other Wikipedia article about the Clique problem in CS it says it is solving the problem for a fixed size k is a problem in P, it can be brute forced in polynomial time.
Can clique be present in G'?
The presence of clique implies that there are K vertices in G, where each of the vertices is connected by an edge with the remaining vertices. This further shows that since these edges are contained in G, therefore they can't be present in G'.

We will show that the Clique problem is NP-complete. It is easy to see that the algorithm above is a polynomial-time algorithm. Putting those facts together, if the algorithm above works, then P = NP. Theorem. The Clique problem is NP-complete. Proof. Lets use abbreviation CP for the Clique problem. IS is the Independent Set problem.

What is an instance of the clique problem?

An instance of the problem is an input specified to the problem. An instance of the Clique problem is a graph G (V, E) and a positive integer K, and the problem is to check whether a clique of size K exists in G.
What is a clique in a graph?
A clique in a graph is a set of vertices where each vertex shares an edge with every other vertex. Thus, a clique in a graph is a subgraph which is a complete graph. An instance of the problem is an input specified to the problem.

Is the $k$-clique problem polynomial time solvable?

This is solvable in polynomial time. Likewise, for any fixed $k$, the $k$-clique problem (given a graph, is there a clique of size $k$) is solvable in polynomial time. But in the clique problem, we are given both the graph and $k$and asked if there is a clique larger than size $k$. This problem is NP-Complete.