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.

翻译：图的隔离问题由Caro和Hansberg于2015年提出，该问题是经典图支配问题的广泛推广，其研究正迅速扩展。本文针对若干自然产生的问题展开探讨。设$F$为一个图，我们证明了当$F$连通时，$F$-隔离集问题是NP完全的。我们研究了图$G$的$F$-隔离数$ι(G,F)$如何受$G$的最小度$d$影响，在$d$充分大的条件下，结合$F$与$G$的阶数，确定了$ι(G,F)$可能最大值的有限范围。此外，借助支配理论及在适当情形下利用Erdos-Posa性质，我们进一步探究了$ι(G,tF)$与$ι(G,F)$的接近程度。