On the correspondence between tree representations of chordal and dually chordal graphs

Chordal graphs and their clique graphs (called dually chordal graphs) possess characteristic tree representations, namely, the clique tree and the compatible tree, respectively. The following problem is studied: given a chordal graph GG, determine if the clique trees of GG are exactly the compatible trees of the clique graph of GG. This leads to a new subclass of chordal graphs, basic chordal graphs, which is here characterized. The question is also approached backwards: given a dually chordal graph GG, we find all the basic chordal graphs with clique graph equal to GG. This approach leads to the possibility of considering several properties of clique trees of chordal graphs and finding their counterparts in compatible trees of dually chordal graphs.