The visible perimeter of an arrangement of disks

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is $O(n^{1/2})$, then there is a stacking order for which the visible perimeter is $\Omega (n^{2/3})$. We also show that this bound cannot be improved in the case of a sufficiently small $n^{1/2}\times n^{1/2}$ uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is $O(n^{3/4})$ with respect to any stacking order. This latter bound cannot be improved either.Finally, we address the case where no more than c disks can have a point in common.These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.