Upper bound for the number of maximal dissociation sets in trees
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912089373671424 |
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| author | Wang, Ziyuan Zhang, Lei Tu, Jianhua Xiong, Liming |
| author_facet | Wang, Ziyuan Zhang, Lei Tu, Jianhua Xiong, Liming |
| contents | Let $G$ be a simple graph. A dissociation set of $G$ is defined as a set of vertices that induces a subgraph in which every vertex has a degree of at most 1. A dissociation set is maximal if it is not contained as a proper subset in any other dissociation set. We introduce the notation $Φ(G)$ to represent the number of maximal dissociation sets in $G$. This study focuses on trees, specifically showing that for any tree $T$ of order $n\geq4$, the following inequality holds:
\[Φ(T)\leq 3^{\frac{n-1}{3}}+\frac{n-1}{3}.\]
We also identify the extremal tree that attains this upper bound. Additionally, to establish the upper bound on the number of maximal dissociation sets in trees of order $n$, we also determine the second largest number of maximal dissociation sets in forests of order $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper bound for the number of maximal dissociation sets in trees Wang, Ziyuan Zhang, Lei Tu, Jianhua Xiong, Liming Combinatorics 05C30, 05C69, 05C05 Let $G$ be a simple graph. A dissociation set of $G$ is defined as a set of vertices that induces a subgraph in which every vertex has a degree of at most 1. A dissociation set is maximal if it is not contained as a proper subset in any other dissociation set. We introduce the notation $Φ(G)$ to represent the number of maximal dissociation sets in $G$. This study focuses on trees, specifically showing that for any tree $T$ of order $n\geq4$, the following inequality holds: \[Φ(T)\leq 3^{\frac{n-1}{3}}+\frac{n-1}{3}.\] We also identify the extremal tree that attains this upper bound. Additionally, to establish the upper bound on the number of maximal dissociation sets in trees of order $n$, we also determine the second largest number of maximal dissociation sets in forests of order $n$. |
| title | Upper bound for the number of maximal dissociation sets in trees |
| topic | Combinatorics 05C30, 05C69, 05C05 |
| url | https://arxiv.org/abs/2410.20462 |