  Finding the exact integrality gap $\alpha$ for the LP relaxation of the
metric Travelling Salesman Problem (TSP) has been an open problem for over
thirty years, with little progress made. It is known that $4/3 \leq \alpha \leq
3/2$, and a famous conjecture states $\alpha = 4/3$. For this problem,
essentially two "fundamental" classes of instances have been proposed. This
fundamental property means that in order to show that the integrality gap is at
most $\rho$ for all instances of metric TSP, it is sufficient to show it only
for the instances in the fundamental class. However, despite the importance and
the simplicity of such classes, no apparent effort has been deployed for
improving the integrality gap bounds for them. In this paper we take a natural
first step in this endeavour, and consider the $1/2$-integer points of one such
class. We successfully improve the upper bound for the integrality gap from
$3/2$ to $10/7$ for a superclass of these points, as well as prove a lower
bound of $4/3$ for the superclass. Our methods involve innovative applications
of tools from combinatorial optimization which have the potential to be more
broadly applied.
