Following up on a previous question: At least as many disks as regions
Given a set $D$ of $n$ same radius disks, embedded in the plane, they induce a number $k$ of connected regions in $\mathbb{R}^2 \setminus \cup_{d \in D}$ .
I am interested in an upper bound on $k$ as a function of $n$.
Does anybody know any good bounds on $k$?
Since the Union Complexity, i.e., the number of arcs on the boundary of $D$ is at most $6n-12$ and each connected region is surrounded by at least 3 disks, it follows that $k \leq 2n - 4$, but I feel that this bound should be much closer to $n$ than to $2n$.