Trees with Given Stability Number and Minimum Number of Stable Sets

TL;DR

This paper characterizes the structure of trees that minimize the number of stable sets given their size and stability number, revealing a star-based partition with specific properties.

Contribution

It introduces a novel structural characterization of extremal trees with minimal stable sets for fixed order and stability number.

Findings

Extremal trees can be partitioned into stars with specific sizes.

Vertices are included in at most two stars, with star centers forming a stable set.

Provides a structural framework for understanding minimal stable set trees.

Abstract

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of size $\lceil \frac{n-1}{n-\alpha} \rceil$ or $\lfloor \frac{n-1}{n-\alpha}\rfloor$, so that every vertex is included in at most two distinct stars, and the centers of these stars form a stable set of the tree.