Comparing trees characteristic to chordal and dually chordal graphs

Chordal and dually chordal graphs possess characteristic tree representations, namely, clique trees and compatible trees, respectively. The following problem is studied: given a chordal graph G, it has to be determined if the clique trees of G are exactly the compatible trees of K(G). This does not always happen. A necessary and sufficient condition so that it is true, in terms of the minimal vertex separators of the graph, is found.