  For a fixed graph $F$, a graph $G$ is said to be $F$-saturated if $G$ does
not contain a subgraph isomorphic to $F$ but does contain $F$ after the
addition of any new edge.
  Let $M_k$ be a matching consisting of $k$ edges and $S_{n,k}$ be the join
graph of a complete graph $K_k$ and an empty graph $\overline{K_{n-k}}$. In
this paper, we prove that for $s \geq3$ and $k\geq 2$, $S_{n,s-2}$ contains the
minimum number of $M_k$ among all $n$-vertex $K_s$-saturated graphs for
sufficiently large $n$, and when $k \leq s-2$, it is the unique extremal graph.
In addition, we also show that $S_{n,1}$ is the unique extremal graph when
$k=2$ and $s=3$.
