Upper bound for the number of maximal dissociation sets in trees

TL;DR

This paper establishes an upper bound on the number of maximal dissociation sets in trees of a given size, identifies the extremal tree achieving this bound, and explores related bounds in forests.

Contribution

It introduces a tight upper bound for the number of maximal dissociation sets in trees and characterizes the extremal trees that attain this bound.

Findings

Upper bound: ^{(n-1)/3} + (n-1)/3 for trees of order na0

Identification of the extremal tree achieving the upper bound

Determination of the second largest number of maximal dissociation sets in forests

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 $n$.