  Let $G$ be a graph and $\mathcal{H}$ be a family of graphs. We say $G$ is
$\mathcal{H}$-saturated if $G$ does not contain a copy of $H$ with
$H\in\mathcal{H}$, but the addition of any edge $e\notin E(G)$ creates at least
one copy of some $H\in\mathcal{H}$ within $G+e$. The saturation number of
$\mathcal{H}$ is the minimum size of an $\mathcal{H}$-saturated graph on $n$
vertices, and the saturation spectrum of $\mathcal{H}$ is the set of all
possible sizes of an $\mathcal{H}$-saturated graph on $n$ vertices. Let
$k\mathcal{C}_{\ge 3}$ be the family of the unions of $k$ vertex-disjoint
cycles. In this note, we completely determine the saturation number and the
saturation spectrum of $k\mathcal{C}_{\ge 3}$ for $k=2$ and give some results
for $k\ge 3$.
