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.

翻译：Djoković-Winkler关系$\Theta$是基于特定顶点距离定义在图边集上的二元关系，在图论中具有重要地位。本文探索了$\Theta$中相对未涉足的"自反对偶补"$\overline\Theta$，其中对于边$e$和$f$，$(e,f)\in \overline\Theta$当且仅当$e=f$或$(e,f)\notin \Theta$。我们建立了$\overline\Theta$与集合$\Delta_{ef}$（包含$e$和$f$顶点间距离）之间的关系，并揭示了其传递闭包$\overline\Theta^*$的复杂性。值得注意的是，我们证明了$\overline\Theta^*$仅在完全多部图的一个受限子类中表现出多个等价类。此外，我们将与$\overline\Theta$一致的非平凡关系$R$刻画为图表示不连通的情况，且每个连通分量是完全图的笛卡尔积（的并）。后者的结果有些出人意料地表明：确定$\overline\Theta^*$无需了解顶点间的距离。进一步地，$\overline\Theta^*$恰好有1个或3个等价类。