In a directed graph $D$, given two distinct vertices $u$ and $v$, the line defined by the ordered pair $(u,v)$ is the set of all vertices $w$ such that $u,v$ and $w$ belong to a shortest directed path in $D$, containing a shortest directed path from $u$ to $v$.

In this work we study the following conjecture: the number of distinct lines in any strongly connected graph is at least its number of vertices, unless there is a line containing all the vertices.

Our main result is that any tournament satisfies this conjecture; we also prove this for bipartite tournaments of diameter at most three.

中文翻译：

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Chen 和 Chvátal 在锦标赛中的猜想

在有向图中 $D$，给定两个不同的顶点 $你$ 和 $v$，由有序对定义的线$(你,v)$ 是所有顶点的集合 $瓦$ 以至于 $你,v$ 和 $瓦$ 属于最短的有向路径 $D$，包含来自的最短有向路径 $你$ 至 $v$.

在这项工作中，我们研究了以下猜想：任何强连通图中不同线的数量至少是它的顶点数，除非有一条线包含所有顶点。

我们的主要结果是任何锦标赛都满足这个猜想；我们也为最多三个直径的双向锦标赛证明了这一点。