关于图形中的完全隔离

Pub Date : 2024-04-25 DOI:10.1007/s00010-024-01057-1

Geoffrey Boyer, Wayne Goddard, Michael A. Henning

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引用次数: 0

摘要

图中的隔离集是这样一个顶点集 S：移除 S 及其邻域不会留下任何边；如果 S 引发了一个没有顶点度为 0 的子图，那么它就是全隔离集。我们证明了大多数图都有一个分割成两个互不相交的全隔离集，并描述了例外情况的特征。利用这一点，我们证明除了 7 循环之外，每个阶为 \(n\ge 4\) 的连通图最多都有一个大小为 n/2 的全孤立集，这是最好的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On total isolation in graphs

An isolating set in a graph is a set S of vertices such that removing S and its neighborhood leaves no edge; it is total isolating if S induces a subgraph with no vertex of degree 0. We show that most graphs have a partition into two disjoint total isolating sets and characterize the exceptions. Using this we show that apart from the 7-cycle, every connected graph of order \(n\ge 4\) has a total isolating set of size at most n/2, which is best possible.

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