  It was recently pointed out that certain SiO$_2$ layer structures and SiO$_2$
nanotubes can be described as full subdivisions aka subdivision graphs of
partial cubes. A key tool for analyzing distance-based topological indices in
molecular graphs is the Djokovi\'c-Winkler relation $\Theta$ and its transitive
closure $\Theta^\ast$. In this paper we study the behavior of $\Theta$ and
$\Theta^\ast$ with respect to full subdivisions. We apply our results to
describe $\Theta^\ast$ in full subdivisions of fullerenes, plane
triangulations, and chordal graphs.
