  Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has
no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in
an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum
number of edges in an $H$-saturated graph on $n$ vertices. K\'aszonyi and Tuza,
in 1986, gave a general upper bound on the saturation number of a graph $H$,
but a nontrivial lower bound has remained elusive. In this paper we give a
general lower bound on $\mathrm{sat}(H,n)$ and prove that it is asymptotically
sharp (up to an additive constant) on a large class of graphs. This class
includes all threshold graphs and many graphs for which the saturation number
was previously determined exactly. Our work thus gives an asymptotic common
generalization of several earlier results. The class also includes disjoint
unions of cliques, allowing us to address an open problem of Faudree, Ferrara,
Gould, and Jacobson.
