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Results on three problems on isolation of graphs

Peter Borg, Yair Caro

TL;DR

It is shown that the F-isolating set problem is NP-complete if F is connected and how close $\iota(G,tF)$ is to $\iota(G,F)$, using domination and, in suitable cases, the Erdos-Posa property.

Abstract

The graph isolation problem was introduced by Caro and Hansberg in 2015. It is a vast generalization of the classical graph domination problem and its study is expanding rapidly. In this paper, we address a number of questions that arise naturally. Let $F$ be a graph. We show that the $F$-isolating set problem is NP-complete if $F$ is connected. We investigate how the $F$-isolation number $ι(G,F)$ of a graph $G$ is affected by the minimum degree $d$ of $G$, establishing a bounded range, in terms of $d$ and the orders of $F$ and $G$, for the largest possible value of $ι(G,F)$ with $d$ sufficiently large. We also investigate how close $ι(G,tF)$ is to $ι(G,F)$, using domination and, in suitable cases, the Erdos-Posa property.

Results on three problems on isolation of graphs

TL;DR

It is shown that the F-isolating set problem is NP-complete if F is connected and how close is to , using domination and, in suitable cases, the Erdos-Posa property.

Abstract

The graph isolation problem was introduced by Caro and Hansberg in 2015. It is a vast generalization of the classical graph domination problem and its study is expanding rapidly. In this paper, we address a number of questions that arise naturally. Let be a graph. We show that the -isolating set problem is NP-complete if is connected. We investigate how the -isolation number of a graph is affected by the minimum degree of , establishing a bounded range, in terms of and the orders of and , for the largest possible value of with sufficiently large. We also investigate how close is to , using domination and, in suitable cases, the Erdos-Posa property.
Paper Structure (4 sections, 16 theorems, 20 equations)

This paper contains 4 sections, 16 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Complexity of $F$-isolation
  3. Bounds on the $F$-isolation number in terms of the minimum degree
  4. Isolation of $t\mathcal{F}$

Key Result

Lemma 1

For any connected graph $F$,

Theorems & Definitions (19)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Corollary 1: CLWX
  • Corollary 2
  • Theorem 3
  • Lemma 3: AW
  • Lemma 4
  • Definition 1
  • ...and 9 more