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Clique (from WikiPedia):

Clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete.

K-Clique problem: Finding a clique of size K. This is NP-complete according to Wiki,

Cliques have also been studied in computer science: finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result many algorithms for finding cliques have been studied.

Let us consider a "constrained k-clique problem" - which is a k-clique problem with a constraint of having a predetermined vertex included in the solution. What would be the complexity of this problem? Is it a known problem in the literature?

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It is still $\mathsf{NP}$-complete. Consider the following reduction from the normal $\mathrm{Clique}$ problem: Given a graph $G$ and some desired clique size $k$, add a new vertex $v^\ast$ which is connected to all vertices of $G$ to obtain a new graph $G^\ast$. Then $(G^\ast, k + 1, v^\ast)$ is a yes-instance of your modified $\mathrm{Clique}$ problem if and only if $(G, k)$ is a yes-instance of $\mathrm{Clique}$.

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