有根有向路图中的小行星
Asteroids in rooted and directed path graphs
摘要：
一个小行星三元组是一个稳定的三个顶点的集合，使得每对顶点都通过一条避开第三个顶点邻域的路径连接。星状三元组在Lekkerkerker和Boland的区间图的经典刻画中起着核心作用。他们的结果说弦图是区间图当且仅当它不包含小行星三元组。本文证明了有向路图的一个类似定理，有向路图是有向树中有向路的交图。为此，我们引入了强路的概念，如果两个不相邻的顶点有一个共同的邻居，或者它们是满足一定条件的两个顶点不相交的无弦路的端点，则它们被强路连接。强小行星三元组是指每对小行星三元组都由强路连接的小行星三元组.我们证明了弦图是有向路图当且仅当它不包含强小行星三元组.我们还介绍了一个相关的概念，asteroidal quadruple，并猜想根路图的一个特征，这是一个根树中的有向路的交集图。
英文摘要：
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.
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