Abstract

In a metric space M = (X, d), a line induced by two distinct points x, x′ ∈ X, denoted by , is the set of points given by

A line is universal whenever .

Chen and Chvátal [Disc. Appl. Math. 156 (2008), 2101-2108.] conjectured that in any finite metric space M = (X, d) either there is a universal line, or there are at least |X| different (nonuniversal) lines. A particular problem derived from this conjecture consists of investigating the properties of M that determine the existence of a universal line, and the problem remains interesting even if we can check that M has at least |X| different lines. Since the vertex set of any connected graph, equipped with the shortest path distance, is a metric space, the problem automatically becomes of interest in graph theory. In this paper, we address the problem of characterizing graphs that have universal lines. We consider several scenarios in which the study can be approached by analysing the existence of such lines in primary subgraphs. We first discuss the wide class of separable graphs, and then describe some particular cases, including those of block graphs, rooted product graphs and corona graphs. We also discuss important classes of nonseparable graphs, including Cartesian product graphs, join graphs and lexicographic product graphs.

摘要

在度量空间M = ( X, d ) 中，由两个不同的点x, x ′ ∈ X导出的线，表示为，是由下式给出的点集

一条线在任何时候都是通用的。

Chen 和 Chvátal [光盘。申请 数学。156 (2008), 2101-2108.] 推测在任何有限度量空间M = ( X , d ) 中要么存在一条通用线，要么至少有 | × | 不同的（非通用的）线路。从这个猜想中导出的一个特定问题包括调查M的属性，这些属性决定了通用线的存在，即使我们可以检查M至少有| ，这个问题仍然很有趣。X| 不同的线路。由于具有最短路径距离的任何连通图的顶点集是一个度量空间，因此该问题自动成为图论中的兴趣所在。在本文中，我们解决了表征具有通用线的图形的问题。我们考虑了几种情况，在这些情况下，可以通过分析主要子图中此类线的存在来进行研究。我们首先讨论广泛的可分离图类别，然后描述一些特殊情况，包括块图、根积图和电晕图。我们还讨论了不可分图的重要类别，包括笛卡尔积图、连接图和字典序积图。