  Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in
$\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a
graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest set $D$
of vertices of $G$ such that the closed neighbourhood of $D$ intersects the
vertex sets of the $\mathcal{F}$-graphs contained by $G$ (equivalently, $G -
N[D]$ contains no $\mathcal{F}$-graph). Thus, $\iota(G,\{K_1\})$ is the
domination number of $G$. For any integer $k \geq 1$, let $\mathcal{F}_{0,k} =
\{K_{1,k}\}$, let $\mathcal{F}_{1,k}$ be the set of regular graphs of degree at
least $k-1$, let $\mathcal{F}_{2,k}$ be the set of graphs whose chromatic
number is at least $k$, and let $\mathcal{F}_{3,k}$ be the union
$\mathcal{F}_{0,k} \cup \mathcal{F}_{1,k} \cup \mathcal{F}_{2,k}$. We prove
that if $G$ is a connected $n$-vertex graph and $\mathcal{F} =
\mathcal{F}_{0,k} \cup \mathcal{F}_{1,k}$, then $\iota(G, \mathcal{F}) \leq
\frac{n}{k+1}$ unless $G$ is a $k$-clique or $k = 2$ and $G$ is a $5$-cycle.
This generalizes a classical bound of Ore on the domination number, a bound of
Caro and Hansberg on the $\{K_{1,k}\}$-isolation number, a bound of the author
on the cycle isolation number, and a bound of Fenech, Kaemawichanurat and the
author on the $k$-clique isolation number. By Brooks' Theorem, the same
inequality holds if $\mathcal{F} = \mathcal{F}_{3,k}$. The bounds are sharp.
