  The modular decomposition of a graph is a canonical representation of its
modules. Algorithms for computing the modular decomposition of directed and
undirected graphs differ significantly, with the undirected case being simpler,
and algorithms for directed graphs often work by reducing the problem to
decomposing undirected graphs. In this paper we show that transitive acyclic
digraphs have the same strong modules as their undirected versions. This
simplifies reduction for transitive digraphs, requiring only the computation of
strongly connected components. Furthermore, we are interested in permutation
graphs, where both the graph and its complement are transitively orientable.
Such graphs may be represented indirectly, as the transitive closure of a given
graph. For non-transitive graphs we present a linear-time algorithm which
allows us to identify prime-free modules w.r.t their transitive closure, which
speeds up both modular decomposition and transitive orientation for sparse
graphs. Finally, we show that any transitive orientation of a digraph's
complement also transitively orients the complement of the digraph's transitive
closure, allowing us to find such orientations in (near-)linear time.
