  Chordal graphs are important in the structural and algorithmic graph theory.
A digraph analogue of chordal graphs was introduced by Haskin and Rose in 1973
but has not been a subject of active studies until recently when a
characterization of semicomplete chordal digraphs in terms of forbidden
subdigraphs was found by Meister and Telle. Locally semicomplete digraphs,
quasi-transitive digraphs, and extended semi-complete digraphs are amongst the
most popular generalizations of semicomplete digraphs. We extend the forbidden
subdigraph characterization of semicomplete chordal digraphs to locally
semicomplete chordal digraphs. We introduce a new class of digraphs, called
weakly quasi-transitive digraphs, which contains quasi-transitive digraphs,
symmetric digraphs, and extended semicomplete digraphs, but is incomparable to
the class of locally semicomplete digraphs. We show that weakly
quasi-transitive digraphs can be recursively constructed by simple
substitutions from transitive oriented graphs, semicomplete digraphs, and
symmetric digraphs. This recursive construction of weakly quasi-transitive
digraphs, similar to the one for quasi-transitive digraphs, demonstrates the
naturalness of the new digraph class. As a by-product, we prove that the
forbidden subdigraphs for semicomplete chordal digraphs are the same for weakly
quasi-transitive chordal digraphs.
