Asteroids in rooted and directed path graphs

TL;DR

This paper characterizes directed path graphs by introducing the concept of strong asteroidal triples, extending classical results about interval graphs and providing a new criterion for identifying these graphs.

Contribution

It proves a characterization theorem for directed path graphs using strong asteroidal triples, analogous to the classical characterization for interval graphs.

Findings

Chordal graphs are directed path graphs iff they contain no strong asteroidal triple.

Introduces the notion of a strong path linking non-adjacent vertices.

Proposes a conjecture for rooted path graphs based on asteroidal quadruples.

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 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.