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The maximum number of maximum dissociation sets in potted graphs

Zejun Huang, Xinwei Zhang

Abstract

A potted graph is a unicyclic graph such that its cycle contains a unique vertex with degree larger than 2. Given a graph $G$, a subset of $V(G)$ is a dissociation set of $G$ if it induces a subgraph with maximum degree at most one. A maximum dissociation set is a dissociation set with maximum cardinality. In this paper, we determine the maximum number of maximum dissociation sets in a potted graph of order $n$ which contains a fixed cycle. Extremal potted graphs attaining this maximum number are also characterized.

The maximum number of maximum dissociation sets in potted graphs

Abstract

A potted graph is a unicyclic graph such that its cycle contains a unique vertex with degree larger than 2. Given a graph , a subset of is a dissociation set of if it induces a subgraph with maximum degree at most one. A maximum dissociation set is a dissociation set with maximum cardinality. In this paper, we determine the maximum number of maximum dissociation sets in a potted graph of order which contains a fixed cycle. Extremal potted graphs attaining this maximum number are also characterized.
Paper Structure (8 sections, 21 theorems, 182 equations, 4 figures, 7 tables)

This paper contains 8 sections, 21 theorems, 182 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the main results
  4. The case that all components of $G-C_k$ have fixed orders
  5. Proof of Theorem \ref{['th1.1']}
  6. Proof of Theorem \ref{['th1.2']}
  7. Proof of Theorem \ref{['th1.3']}
  8. Conclusion

Key Result

Theorem 1.1

Let $n$, $k$ be positive integers such that $n\ge k+4$ and $k\equiv0~(\bmod~3)$. If $G\in\mathcal{P}(n,k)$, then with equality if and only if

Figures (4)

  • Figure 1: The families $\mathcal{T}_1(w),\ldots,\mathcal{T}_5(w)$ and $\mathcal{G}_1,\ldots,\mathcal{G}_{9}$
  • Figure 2: Trees in the family $\mathcal{H}$
  • Figure 3: The potted graphs of order $n\le k+3$
  • Figure 4: Seven labeled trees $T$ of order $n$

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 22 more