  In this article we deal with the problems of finding the disimplicial arcs of
a digraph and recognizing some interesting graph classes defined by their
existence. A diclique of a digraph is a pair $V \to W$ of sets of vertices such
that $v \to w$ is an arc for every $v \in V$ and $w \in W$. An arc $v \to w$ is
disimplicial when $N^-(w) \to N^+(v)$ is a diclique. We show that the problem
of finding the disimplicial arcs is equivalent, in terms of time and space
complexity, to that of locating the transitive vertices. As a result, an
efficient algorithm to find the bisimplicial edges of bipartite graphs is
obtained. Then, we develop simple algorithms to build disimplicial elimination
schemes, which can be used to generate bisimplicial elimination schemes for
bipartite graphs. Finally, we study two classes related to perfect disimplicial
elimination digraphs, namely weakly diclique irreducible digraphs and diclique
irreducible digraphs. The former class is associated to finite posets, while
the latter corresponds to dedekind complete finite posets.
