Abstract

The set of vertices of a graph is called distance- \( k \) independent if the distance between any two of its vertices is greater than some integer \( k \geq 1 \). In this paper, we describe \( n \)-vertex trees with a given diameter \( d \) that have the maximum and minimum possible number of distance- \( k \) independent sets among all such trees. The maximum problem is solvable for the case of \( 1 < k < d \leq 5 \). The minimum problem is much simpler and can be solved for all \( 1 < k < d < n \).

中文翻译：

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给定直径和距离极值数的树 - $$k$$ 独立集

摘要

如果图的任意两个顶点之间的距离大于某个整数 \( k \geq 1 \) ，则图的顶点集被称为距离\( k \) 独立。在本文中，我们描述了 具有给定直径 \(d\ )的\(n\) -顶点树，它们在所有此类树中具有最大和最小可能数量的距离 -\(k\)独立集。对于\( 1 < k < d \leq 5 \)的情况，最大问题是可解的 。最小问题要简单得多，并且可以解决所有 \( 1 < k < d < n \)。