Trees with Given Stability Number and Minimum Number of Stable Sets

TL;DR

The main result is that the edges of a non-trivial extremal tree can be partitioned into n − α stars, each of size n-1 n-alpha, 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.

Abstract

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $α$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-α$ stars, each of size $\lceil \frac{n-1}{n-α} \rceil$ or $\lfloor \frac{n-1}{n-α}\rfloor$, 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.

Figures (10)

• Figure 1: An extremal tree for $n= 18$ and $\alpha = 13$. White vertices are the centers of the stars.
• Figure 2: A tree of stars. The white vertices are the centers of the tree.
• Figure 3: A tree which is almost a tree of stars. Its exposed center is drawn in grey.
• Figure 4: A rotation $\rho=(yx, yx')$.
• Figure 5: Rotations $\rho_1 = (v'v, v'w)$ and $\rho_2 = (vv_1, vw)$.

Theorems & Definitions (34)

• Lemma 1
• Lemma 2
• proof
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof : Proof of Lemma \ref{['lem-ToS-center']}
• Lemma 7
• proof