  We study a random graph $G$ with given degree sequence $\boldsymbol{d}$, with
the aim of characterising the degree sequence of the subgraph induced on a
given set $S$ of vertices. For suitable $\boldsymbol{d}$ and $S$, we show that
the degree sequence of the subgraph induced on $S$ is essentially concentrated
around a sequence that we can deterministically describe in terms of
$\boldsymbol{d}$ and $S$. We then give an application of this result,
determining a threshold for when this induced subgraph contains a giant
component. We also apply a similar analysis to the case where $S$ is chosen by
randomly sampling vertices with some probability $p$, i.e. site percolation,
and determine a threshold for the existence of a giant component in this model.
We consider the case where the density of the subgraph is either constant or
slowly going to $0$ as $n$ goes to infinity, and the degree sequence
$\boldsymbol{d}$ of the whole graph satisfies a certain maximum degree
condition. Analogously, in the percolation model we consider the cases where
either $p$ is a constant or where $p \to 0$ slowly. This is similar to work of
Fountoulakis in 2007 and Janson in 2009, but we work directly in the random
graph model to avoid the limitations of the configuration model that they used.
