On Trees with a Given Diameter and the Extremal Number of Distance-\(k\) Independent Sets

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 \( 1 < k < d < n \).