  Chordal graphs are important in algorithmic graph theory. Chordal digraphs
are a digraph analogue of chordal graphs and have been a subject of active
studies recently. Unlike chordal graphs, chordal digraphs lack many structural
properties such as forbidden subdigraph or representation characterizations. In
this paper we introduce the notion of semi-strict chordal digraphs which form a
class strictly between chordal digraphs and chordal graphs. Semi-strict chordal
digraphs have rich structural properties. We characterize semi-strict chordal
digraphs in terms of knotting graphs, a notion analogous to the one introduced
by Gallai for the study of comparability graphs. We also give forbidden
subdigraph characterizations of semi-strict chordal digraphs within the cases
of locally semicomplete digraphs and weakly quasi-transitive digraphs.
