Characterizing directed path graphs by forbidden asteroids |
| |
Authors: | Kathie Cameron Chính T Hoàng Benjamin Lévêque |
| |
Institution: | 1. Department Of Mathematics, Wilfrid Laurier University Waterloo, Ont., Canada N2L 3C5;2. Department Of Physics And Computer Science Wilfrid Laurier University, Waterloo, Ont., Canada N2L 3C5;3. Cnrs, Lirmm, 161 Rue Ada 34392 Montpellier Cedex 05, France |
| |
Abstract: | 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 does not contain an 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 special connection. Two non‐adjacent vertices are linked by a special connection if either they have a common neighbor or they are the endpoints of two vertex‐disjoint chordless paths satisfying certain conditions. A special asteroidal triple is an asteroidal triple such that each pair is linked by a special connection. We prove that a chordal graph is a directed path graph if and only if it does not contain a special asteroidal triple. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:103‐112, 2011 |
| |
Keywords: | intersection graph directed path graph asteroidal triple |
|
|