Upper bound for the number of maximal dissociation sets in trees

Ziyuan Wang, Lei Zhang, Jianhua Tu*, Liming Xiong

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 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≥4, the following inequality holds: Φ(T)≤3 [Formula presented]. We also identify extremal trees that attain 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.

Original languageEnglish
Article number114545
JournalDiscrete Mathematics
Volume348
Issue number9
DOIs
Publication statusPublished - Sept 2025
Externally publishedYes

Keywords

  • Enumeration
  • Forests
  • Maximal dissociation sets
  • Trees

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