  The Djokovi\'{c}-Winkler relation $\Theta$ is a binary relation defined on
the edge set of a given graph that is based on the distances of certain
vertices and which plays a prominent role in graph theory. In this paper, we
explore the relatively uncharted ``reflexive complement'' $\overline\Theta$ of
$\Theta$, where $(e,f)\in \overline\Theta$ if and only if $e=f$ or $(e,f)\notin
\Theta$ for edges $e$ and $f$. We establish the relationship between
$\overline\Theta$ and the set $\Delta_{ef}$, comprising the distances between
the vertices of $e$ and $f$ and shed some light on the intricacies of its
transitive closure $\overline\Theta^*$. Notably, we demonstrate that
$\overline\Theta^*$ exhibits multiple equivalence classes only within a
restricted subclass of complete multipartite graphs. In addition, we
characterize non-trivial relations $R$ that coincide with $\overline\Theta$ as
those where the graph representation is disconnected, with each connected
component being the (join of) Cartesian product of complete graphs. The latter
results imply, somewhat surprisingly, that knowledge about the distances
between vertices is not required to determine $\overline\Theta^*$. Moreover,
$\overline\Theta^*$ has either exactly one or three equivalence classes.
