Chordality of locally semicomplete and weakly quasi-transitive digraphs

TL;DR

This paper extends the understanding of chordal digraphs by characterizing locally semicomplete and introducing weakly quasi-transitive digraphs, revealing their recursive structure and forbidden subdigraphs.

Contribution

It generalizes forbidden subdigraph characterizations to broader classes and introduces weakly quasi-transitive digraphs with a recursive construction.

Findings

Forbidden subdigraphs for semicomplete chordal digraphs apply to weakly quasi-transitive chordal digraphs.

Weakly quasi-transitive digraphs can be built from transitive, semicomplete, and symmetric digraphs.

The class of weakly quasi-transitive digraphs is structurally natural and broad.

Abstract

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.