  Given a set of $n$ disks of radius $R$ in the Euclidean plane, the Traveling
Salesman Problem With Neighborhoods (TSPN) on uniform disks asks for the
shortest tour that visits all of the disks. The problem is a generalization of
the classical Traveling Salesman Problem(TSP) on points and has been widely
studied in the literature. For the case of disjoint uniform disks of radius
$R$, Dumitrescu and Mitchell[2001] show that the optimal TSP tour on the
centers of the disks is a $3.547$-approximation to the TSPN version. The core
of their analysis is based on bounding the detour that the optimal TSPN tour
has to make in order to visit the centers of each disk and shows that it is at
most $2Rn$ in the worst case. H\"{a}me, Hyyti\"{a} and Hakula[2011] asked
whether this bound is tight when $R$ is small and conjectured that it is at
most $\sqrt{3}Rn$.
  We further investigate this question and derive structural properties of the
optimal TSPN tour to describe the cases in which the bound is smaller than
$2Rn$. Specifically, we show that if the optimal TSPN tour is not a straight
line, at least one of the following is guaranteed to be true: the bound is
smaller than $1.999Rn$ or the $TSP$ on the centers is a $2$-approximation. The
latter bound of $2$ is the best that we can get in general. Our framework is
based on using the optimality of the TSPN tour to identify local structures for
which the detour is large and then using their geometry to derive better lower
bounds on the length of the TSPN tour. This leads to an improved approximation
factor of $3.53$ for disjoint uniform disks and $6.728$ for the general case.
We further show that the H\"{a}me, Hyyti\"{a} and Hakula conjecture is true for
the case of three disks and discuss the method used to obtain it. Along the
way, we will uncover one way in which the TSPN problem is structurally
different from the classical TSP.
