  Given a convex $n$-gon, we can draw $n$ disks (called side disks) where each
disk has a different side of the polygon as diameter and the midpoint of the
side as its center. The intersection graph of such disks is the undirected
graph with vertices the $n$ disks and two disks are adjacent if and only if
they have a point in common. Such a graph was introduced by Huemer and
P\'erez-Lantero in 2016, proved to be planar and Hamiltonian. In this paper we
study further combinatorial properties of this graph. We prove that the
treewidth is at most 3, by showing an $O(n)$-time algorithm that builds a tree
decomposition of width at most 3, given the polygon as input. This implies that
we can construct the intersection graph of the side disks in $O(n)$ time. We
further study the independence number of this graph, which is the maximum
number of pairwise disjoint disks. The planarity condition implies that for
every convex $n$-gon we can select at least $\lceil n/4 \rceil$ pairwise
disjoint disks, and we prove that for every $n\ge 3$ there exist convex
$n$-gons in which we cannot select more than this number. Finally, we show that
our class of graphs includes all outerplanar Hamiltonian graphs except the
cycle of length four, and that it is a proper subclass of the planar
Hamiltonian graphs.
