  The $\ell$-component connectivity (or $\ell$-connectivity for short) of a
graph $G$, denoted by $\kappa_\ell(G)$, is the minimum number of vertices whose
removal from $G$ results in a disconnected graph with at least $\ell$
components or a graph with fewer than $\ell$ vertices. This generalization is a
natural extension of the classical connectivity defined in term of minimum
vertex-cut. As an application, the $\ell$-connectivity can be used to assess
the vulnerability of a graph corresponding to the underlying topology of an
interconnection network, and thus is an important issue for reliability and
fault tolerance of the network. So far, only a little knowledge of results have
been known on $\ell$-connectivity for particular classes of graphs and small
$\ell$'s. In a previous work, we studied the $\ell$-connectivity on
$n$-dimensional alternating group networks $AN_n$ and obtained the result
$\kappa_3(AN_n)=2n-3$ for $n\geqslant 4$. In this sequel, we continue the work
and show that $\kappa_4(AN_n)=3n-6$ for $n\geqslant 4$.
