The maximum number of maximum dissociation sets in trees

TL;DR

This paper determines the maximum number of maximum dissociation sets in trees of a given order and characterizes all extremal trees, extending previous work on maximum independent sets.

Contribution

It establishes the maximum number of maximum dissociation sets in trees and provides structural descriptions of extremal trees, inspired by prior results on independent sets.

Findings

Maximum number of maximum dissociation sets in trees of order n is given by a specific formula depending on n mod 3.

Complete structural descriptions of all extremal trees are provided.

The results extend known bounds from maximum independent sets to maximum dissociation sets.

Abstract

A subset of vertices is a {\it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {\it maximum dissociation set} if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. Zito [J. Graph Theory {\bf 15} (1991) 207--221] proved that the maximum number of maximum independent sets of a tree of order $n$ is $2^{\frac{n-3}{2}}$ if $n$ is odd, and $2^{\frac{n-2}{2}}+1$ if $n$ is even and also characterized all extremal trees with the most maximum independent sets, which solved a question posed by Wilf. Inspired by the results of Zito, in this paper, by establishing four structure theorems and a result of $k$-K\"{o}nig-Egerv\'{a}ry graph, we show that the maximum number of maximum dissociation sets in a tree of order $n$ is \begin{center} $\left\{ \begin{array}{ll} 3^{\frac{n}{3}-1}+\frac{n}{3}+1, & \hbox{if $n\equiv0\pmod{3}$;} 3^{\frac{n-1}{3}-1}+1, & \hbox{if $n\equiv1\pmod{3}$;} 3^{\frac{n-2}{3}-1}, & \hbox{if $n\equiv2\pmod{3}$,} \end{array} \right.$ \end{center} and also give complete structural descriptions of all extremal trees on which these maxima are achieved.