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.