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Structure of tight (k,0)-stable graphs

Dingding Dong, Sammy Luo

Abstract

We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals $\lfloor{\frac{n-k+1}{2}\rfloor}+\ell$, the maximum possible, where $n$ is the vertex number of G. Answering a question of Dong and Wu, we show that every tight $(2,0)$-stable graph with odd vertex number must be an odd cycle. Moreover, we show that for all $k\geq 3$, every tight $(k,0)$-stable graph has at most $k+6$ vertices.

Structure of tight (k,0)-stable graphs

Abstract

We say that a graph G is -stable if removing vertices from it reduces its independence number by at most . We say that G is tight -stable if it is -stable and its independence number equals , the maximum possible, where is the vertex number of G. Answering a question of Dong and Wu, we show that every tight -stable graph with odd vertex number must be an odd cycle. Moreover, we show that for all , every tight -stable graph has at most vertices.
Paper Structure (4 sections, 6 theorems, 6 equations, 2 figures)

This paper contains 4 sections, 6 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of \ref{['thm:main-1']}(a)--(c)
  3. Proof of \ref{['thm:main-1']}(d) and \ref{['thm:main-2']}
  4. Further questions

Key Result

Theorem 1.1

Let $G$ be a tight $(k,0)$-stable graph on $n$ vertices.

Figures (2)

  • Figure 1: Some canonical cases in \ref{['thm:main-1']}(b)
  • Figure :

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • proof : Proof of \ref{['thm:main-1']}(a)
  • proof : Proof of \ref{['thm:main-1']}(b)
  • proof : Proof of \ref{['thm:main-1']}(c)
  • Definition 3.1
  • ...and 5 more