Clique trees of chordal graphs: leafage and 3-asteroidals

Chordal graphs were characterized as those graphs having a tree, called clique tree, whose vertices are the cliques of the graph and for every vertex in the graph, the set of cliques that contain it form a subtree of clique tree. In this work, we study the relationship between the clique trees of a chordal graph and its subgraphs. We will prove that clique trees can be described locally and all clique trees of a graph can be obtained from clique trees of subgraphs. In particular, we study the leafage of chordal graphs, that is the minimum number of leaves among the clique trees of the graph. It is known that interval graphs are chordal graphs without 3-asteroidals. We will prove a generalization of this result using the framework developed in the present article. We prove that in a clique tree that realizes the leafage, for every vertex of degree at least 3, and every choice of 3 branches incident to it, there is a 3−asteroidal in these branches.