Extended star graphs

Chordal graphs, which are intersection graph of subtrees of a tree, can be represented on trees. Some representation of a chordal graph often reduces the size of the data structure needed to store the graph, permitting the use of extremely efficient algorithms that take advantage of the compactness of the representation. An extended star graph is the intersection graph of a family of subtrees of a tree that has exactly one vertex of degree at least three. An asteroidal triple in a graph is a set of three non-adjacent vertices such that for any two of them there exists a path between them that does not intersect the neighborhood of the third. Several subclasses of chordal graphs (interval graphs, directed path graphs) have been characterized by forbidden asteroids. In this paper, we define, a subclass of chordal graphs, called extended star graphs, prove a characterization of this class by forbidden asteroids and show open problems.