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The minimum number of maximal dissociation sets in unicyclic graphs

Junxia Zhang, Xiangyu Ren, Maoqun Wang

Abstract

A subset of vertices in a graph $G$ is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, it is shown that for $n\geq 3$, every unicyclic graph contains a minimum of $\lfloor n/2\rfloor+2$ maximal dissociation sets. We also show the graphs that attain this minimum bound.

The minimum number of maximal dissociation sets in unicyclic graphs

Abstract

A subset of vertices in a graph is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, it is shown that for , every unicyclic graph contains a minimum of maximal dissociation sets. We also show the graphs that attain this minimum bound.

Paper Structure

This paper contains 5 sections, 11 theorems, 31 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th1']}
  4. Date Availability
  5. Statements and Declarations

Key Result

Theorem 1.1

Let $G$ be a unicyclic graph of order $n$ ($n\geq 3$), we have $\phi(G)\geq \lfloor \frac{n}{2}\rfloor+2$, with equality if and only if one of the following statements holds: (1) $n$ is odd, $G=U\left(\frac{n-3}{2},\frac{n-3}{2}\right)$; (2) $n$ is even, $G=U\left(\frac{n-2}{2},\frac{n-4}{2}\right)$

Figures (6)

  • Figure 1: $U_{6,0},U_{5,1}$ and $U_{4,4}$.
  • Figure 2: The unicyclic graphs $U\left(\frac{n-3}{2},\frac{n-3}{2}\right)$ and $U\left(\frac{n-2}{2},\frac{n-4}{2}\right)$.
  • Figure 3: The trees $T\left(\frac{n}{2},\frac{n-2}{2}\right)$, $T\left(\frac{n-1}{2},\frac{n-1}{2}\right)$ and $T\left(\frac{n+1}{2},\frac{n-3}{2}\right)$.
  • Figure 4:
  • Figure 5: The unicyclic graphs $G_1$ and $G_2$.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Lemma 3.1
  • proof
  • ...and 6 more