Asteroids in rooted and directed path graphs

Kathie Cameron (1) , Chinh T. Hoàng (1) , Benjamin Lévêque (2)

1 WLU - Wilfrid Laurier University
2 ALGCO - Algorithmes, Graphes et Combinatoire

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.

Mots clés: Intersection graph; Directed path graph; Asteroidal triple

Domaines: Mathématique discrète [cs.DM]

Kathie Cameron, Chinh T. Hoàng, Benjamin Lévêque. Asteroids in rooted and directed path graphs. Algorithmic Graph Theory, Centre for Discrete Mathematics and its applications (DIMAP), May 2009, Warwick, United Kingdom. pp.67-74. ⟨hal-00347163v2⟩