Characterizing directed path graphs by forbidden asteroids

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 103104 JOURNAL OF GRAPH THEORY 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. ᭧

On Models of Directed Path Graphs Non Rooted Directed Path Graphs

“…Special connection linking vertices a 1 and a 2 in a graph were defined in [2] in order to characterize directed path graph by forbidden asteroidals. Here, we will need only two types of these connections: 1 , a 2 , a, b, c, d and edges a 1 a, a 1 b, a 2 c, a 2 d, ab, bc, cd, T (a 1 , a 2 ) is a directed path.…”

“…of directed subpaths of a directed tree. Panda [12], found the characterization of directed path graphs by forbidden induced subgraphs and then Cameron, Hoáng and Lévêque [2] gave a characterization of this class in terms of forbidden asteroidal triples. Secondly, a graph is a rooted path graph if it is the intersection graph of a family of directed subpaths of a rooted tree.…”

Path Graphs, Clique Trees, and Flowers

“…Interestingly, both Panda and Lévêque et al [11,14] list the forbidden subgraphs of directed path and path graphs respectively by direct proofs, leaving the characterization of these classes in terms of asteroidal triples open. For directed path graphs, this problem was resolved by Cameron et al [1], who gave a comparable theorem to that of Lekkerkerker and Boland: Theorem 1.2. A chordal graph G is a directed path graph if and only if it does not contain a special asteroidal triple.…”

“…A special asteroidal triple is an asteroidal triple where each pair of vertices in the triple must be connected by some special subgraph: see Section 4 for a further discussion of this result. Notably, a characterization for path graphs via asteroidal triples of this type was explicitly left open by both [1,11].…”

Special asteroidal quadruple on directed path graph non rooted path graph

“…Special asteroidal triple in a graph G is an asteroidal triple such that each pair is linked by a special connection. A special asteroidal triples play a central role in a characterization of directed path graphs by Cameron, Hoáng and Lévêque [2]. They also introduce a related notion of asteroidal quadruple and conjecture a characterization of rooted path graphs [1].…”