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arXiv · 2410.20462

Upper bound for the number of maximal dissociation sets in trees

Ziyuan Wang·Lei Zhang·Jianhua Tu·Liming Xiong
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Abstract

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 $\Phi(G)$ 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: \[\Phi(T)\leq 3^{\frac{n-1}{3}}+\frac{n-1}{3}.\] 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 $n$.

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Ziyuan Wang, Lei Zhang, Jianhua Tu, Liming Xiong. 2024-10-27. Upper bound for the number of maximal dissociation sets in trees. https://arxiv.org/abs/2410.20462

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