  An asteroidal triple is a stable set of three vertices such that each pair is
connected by a path avoiding the neighborhood of the third vertex. Asteroidal
triples play a central role in a classical characterization of interval graphs
by Lekkerkerker and Boland. Their result says that a chordal graph is an
interval graph if and only if it contains no asteroidal triple. In this paper,
we prove an analogous theorem for directed path graphs which are the
intersection graphs of directed paths in a directed tree. For this purpose, we
introduce the notion of a strong path. Two non-adjacent vertices are linked by
a strong path if either they have a common neighbor or they are the endpoints
of two vertex-disjoint chordless paths satisfying certain conditions. A strong
asteroidal triple is an asteroidal triple such that each pair is linked by a
strong path. We prove that a chordal graph is a directed path graph if and only
if it contains no strong asteroidal triple. We also introduce a related notion
of asteroidal quadruple, and conjecture a characterization of rooted path
graphs which are the intersection graphs of directed paths in a rooted tree.
