I posted this question on math.stackexchange, but it might be more suited for this (research level) website:

Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces 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 (a reference for) a good upperbound on $k$?

Since the Union Complexity, i.e., the number of arcs on the boundary of $D$ is at most $6n-12$ (if $n \geq 3$) and each connected region is bounded 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 $2n$.

Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces 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 (a reference for) a good upperbound on $k$?

Since the Union Complexity, i.e., the number of arcs on the boundary of $D$ is at most $6n-12$ (if $n \geq 3$) and each connected region is bounded 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 $2n$.

I posted this question on math.stackexchange, but it might be more suited for this (research level) website:

Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces 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 $6n-12$ (if $n \geq 3$) and each connected region is bounded 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 $2n$.

I posted this question on math.stackexchange, but it might be more suited for this (research level) website:

Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces 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 (a reference for) a good upperbound on $k$?

Since the Union Complexity, i.e., the number of arcs on the boundary of $D$ is at most $6n-12$ (if $n \geq 3$) and each connected region is bounded 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 $2n$.