On basic chordal graphs and some of its subclasses

Abstract

Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. In this work, we consider some subclasses of basic chordal graphs. One of them is the class of hereditary basic chordal graphs, which will turn out to have many possible characterizations. Those characterizations will show that the class was already studied, but under different names and in different contexts. We also study the connection between basic chordal graphs and some subclasses of chordal graphs with special clique trees, like DV graphs and RDV graphs. As a result, it will be possible to define the classes of basic DV graphs and basic RDV graphs. Additionally, we study the behavior of the clique operator over all the considered subclasses.