  Seminal works on light spanners over the years provide spanners with optimal
lightness in various graph classes, such as in general graphs, Euclidean
spanners, and minor-free graphs. Three shortcomings of previous works on light
spanners are: (1) The techniques are ad hoc per graph class, and thus can't be
applied broadly. (2) The runtimes of these constructions are almost always
sub-optimal, and usually far from optimal. (3) These constructions are optimal
in the standard and crude sense, but not in a refined sense that takes into
account a wider range of involved parameters.
  This work aims at addressing these shortcomings by presenting a unified
framework of light spanners in a variety of graph classes. Informally, the
framework boils down to a transformation from sparse spanners to light
spanners; since the state-of-the-art for sparse spanners is much more advanced
than that for light spanners, such a transformation is powerful. Our framework
is developed in two papers. The current paper is the second of the two -- it
builds on the basis of the unified framework laid in the first paper, and then
strengthens it to achieve more refined optimality bounds for several graph
classes. Among various applications and implications of our framework, we
highlight here the following:
  For $K_r$-minor-free graphs, we provide a $(1+\epsilon)$-spanner with
lightness $\tilde{O}_{r,\epsilon}( \frac{r}{\epsilon} + \frac{1}{\epsilon^2})$,
improving the lightness bound $\tilde{O}_{r,\epsilon}( \frac{r}{\epsilon^3})$
of Borradaile, Le and Wulff-Nilsen. We complement our upper bound with a lower
bound construction, for which any $(1+\epsilon)$-spanner must have lightness
$\Omega(\frac{r}{\epsilon} + \frac{1}{\epsilon^2})$. We note that the quadratic
dependency on $1/\epsilon$ we proved here is surprising, as the prior work
suggested that the dependency on $\epsilon$ should be $1/\epsilon$.
