Modular decomposition of transitive graphs and transitively orienting their complements

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.