  For an integer $\ell\geqslant 2$, the $\ell$-component connectivity 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 is a natural
generalization of the classical connectivity of graphs defined in term of the
minimum vertex-cut and is a good measure of robustness for the graph
corresponding to a network. So far, the exact values of $\ell$-connectivity are
known only for a few classes of networks and small $\ell$'s. It has been
pointed out in~[Component connectivity of the hypercubes, Int. J. Comput. Math.
89 (2012) 137--145] that determining $\ell$-connectivity is still unsolved for
most interconnection networks, such as alternating group graphs and star
graphs. In this paper, by exploring the combinatorial properties and
fault-tolerance of the alternating group graphs $AG_n$ and a variation of the
star graphs called split-stars $S_n^2$, we study their $\ell$-component
connectivities. We obtain the following results: (i) $\kappa_3(AG_n)=4n-10$ and
$\kappa_4(AG_n)=6n-16$ for $n\geqslant 4$, and $\kappa_5(AG_n)=8n-24$ for
$n\geqslant 5$; (ii) $\kappa_3(S_n^2)=4n-8$, $\kappa_4(S_n^2)=6n-14$, and
$\kappa_5(S_n^2)=8n-20$ for $n\geqslant 4$.
