关于有向图连通性的说明

arXiv - MATH - Combinatorics Pub Date : 2024-09-18 DOI:arxiv-2409.12137

Stelios Stylianou

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引用次数: 0

摘要

如果删除任何一条边都会减少不同顶点$(u,v)$的有序对的数量，从而存在一条从$u$到$v$的有向路径，那么我们就说在$n$顶点上的有向图$G$是不冗余的。对于 \mathbb{N}$ 中的每一个 $n ，我们确定了这样一个图可能具有的最大边数。这就以强的形式解决了克莱恩和罗素的问题。

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A note on connectivity in directed graphs

We say a directed graph $G$ on $n$ vertices is irredundant if the removal of any edge reduces the number of ordered pairs of distinct vertices $(u,v)$ such that there exists a directed path from $u$ to $v$. We determine the maximum possible number of edges such a graph can have, for every $n \in \mathbb{N}$. We also characterize the cases of equality. This resolves, in a strong form, a question of Crane and Russell.

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arXiv - MATH - Combinatorics

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