Trees with Given Stability Number and Minimum Number of Stable Sets
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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2010
- DOI:
- 10.48550/arXiv.1002.1270
- arXiv:
- arXiv:1002.1270
- Bibcode:
- 2010arXiv1002.1270B
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- v2: Referees' comments incorporated