Chordal digraphs

We re-consider perfect elimination digraphs, that were introduced by Haskins and Rose in 1973, and view these graphs as directed analogues of chordal graphs. Several structural properties of chordal graphs that are crucial for algorithmic applications carry over to the directed setting, including notions like simplicial vertices, perfect elimination orderings, and vertex layouts. We show that semi-complete perfect elimination digraphs are also characterised by a set of forbidden induced subgraphs resemblant of chordless cycles. Moreover, just as the chordal graphs are related to treewidth, the perfect elimination digraphs are related to Kelly-width.