# Computing distance to clique in FPT time

I am a research scholar, and I currently work in parameterized algorithms. My current work involves proving that a problem is FPT for the parameter distance to clique. Although it is known that computing a distance to clique of size $$k$$ can be done in FPT time using a simple branching algorithm, I couldn't figure out how. Please elaborate on this. Once I get this, then I can assume that the distance to clique is given and solve the rest.
For a graph $$G = (V, E)$$, the parameter distance to clique is the cardinality of the smallest set $$D \subseteq V$$ such that $$V \setminus D$$ is a clique.

• If the distance to clique is 0, then its about solving the problem for a clique. If i assume that for distance to clique it is FPT, then in this case it is polynomial time solvable on clique. @PålGD Jan 18, 2023 at 13:10

Let us take the complement graph $$\overline G$$ of $$G$$. Now, we instead want to find the fewest number of vertices to remove such that the resulting graph is edgeless.
Recall that if $$I$$ is edgeless (i.e. an independent set), then $$C$$ is a vertex cover of $$G$$. Hence, distance to clique is simply vertex cover in the complement graph.