# Maximal subsets of a point set which fit in a unit disk

Suppose that there are a set $$P$$ of $$n$$ points on the plane, and let $$P_1, \dots, P_k$$ be not necessarily disjoint subsets of $$P$$ such that every point in $$P_i|\ 1 \leq i \leq k$$ fits inside a unit disk $$D_i$$.

Moreover, each $$P_i$$ is maximal. This means that if the corresponding unit disk $$D_i$$ moves to cover another point, then one point which was inside the disk will be uncovered.

Here is an example:

In the above figure, there are three maximal subsets.

I don’t know whether this problem has a name or was studied before, but my question is:

1. Can $$k$$ be $$O(1)^n$$?
2. If not, then can we find those subsets in polynomial time w.r.t. $$n$$?