On the maximum number of maximum dissociation sets in trees with given dissociation number

TL;DR

This paper determines the maximum number of maximum dissociation sets in trees with a given dissociation number and identifies the extremal trees that achieve this maximum.

Contribution

It introduces a new extremal problem in graph theory, specifically for dissociation sets in trees, and provides exact maximum counts and characterizations.

Findings

Maximum number of maximum dissociation sets in trees is established.

Extremal trees achieving the maximum are characterized.

The results advance understanding of dissociation set distributions in trees.

Abstract

In a graph $G$, a subset of vertices is a dissociation set if it induces a subgraph with vertex degree at most 1. A maximum dissociation set is a dissociation set of maximum cardinality. The dissociation number of $G$, denoted by $\psi(G)$, is the cardinality of a maximum dissociation set of $G$. Extremal problems involving counting the number of a given type of substructure in a graph have been a hot topic of study in extremal graph theory throughout the last few decades. In this paper, we determine the maximum number of maximum dissociation sets in a tree with prescribed dissociation number and the extremal trees achieving this maximum value.