Results on three problems on isolation of graphs

TL;DR

This paper explores the computational complexity, degree-based bounds, and relationships of the graph isolation problem, a generalization of domination, with new NP-completeness results and bounds related to graph degree and subgraph structures.

Contribution

It establishes NP-completeness for the $F$-isolating set problem when $F$ is connected, and provides bounds on the $F$-isolation number based on minimum degree and subgraph properties.

Findings

NP-completeness of $F$-isolating set problem for connected $F$

Bounds on $(G,F)$ in terms of minimum degree $d$

Analysis of $(G,tF)$ relative to $(G,F)$ using domination and Erdos-Posa

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 $\iota(G,F)$ 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 $\iota(G,F)$ with $d$ sufficiently large. We also investigate how close $\iota(G,tF)$ is to $\iota(G,F)$, using domination and, in suitable cases, the Erdos-Posa property.