The rectangle-partition number and the number of horizontral edges

Question

The rectangle-partition-number of a rectilinear polygon $P$ is the smallest number of pairwise-disjoint axis-parallel rectangles required to cover $P$. Some examples:

(in the last example, $P$ is actually a pair of disjoint polygons).

Let $R(n)$ be the largest rectangle-partition-number of a rectilinear polygon with $n$ horizontal sides. The figure below shows that $R(2)\geq 1$, $R(4)\geq 3$, $R(6)\geq 5$ and $R(8)\geq 6$.

What is an upper bound on $R(n)$?

Answer 1

I found an almost complete answer. A theorem cited by Eppstein 2009, page 4 says that:

the minimum number of rectangles in a partition of a polygon with $v$ vertices and $h$ holes is $v/2 + h − g − 1$, where $g$ is the maximum size of a set of disjoint good diagonals.

The number of horizontal sides is exactly $n=v/2$. So, if the polygon is hole-free, an upper bound on $R(n)$ is: $R(n)\leq n-1$.