  For a family $\mathcal{F}$ of graphs, $sat(n,\mathcal{F})$ is the minimum
number of edges in a graph $G$ on $n$ vertices which does not contain any of
the graphs in $\mathcal{F}$ but such that adding any new edge to $G$ creates a
graph in $\mathcal{F}$. For singleton families $\mathcal{F}$, Tuza conjectured
that $sat(n,\mathcal{F})/n$ converges and Truszczynski and Tuza discovered that
either $sat(n,\mathcal{F})= \left(1-1/r\right)n+o(n)$ for some integer $r \geq
1$ or $ sat(n,\mathcal{F}) \geq n+o(n) $. This is often cited in the literature
as the main progress towards proving Tuza's Conjecture. Unfortunately, the
proof is flawed. We give a correct proof, which requires a novel construction.
Moreover, for finite families $\mathcal{F}$, we completely determine the
possible asymptotic behaviours of $sat(n,\mathcal{F})$ in the sparse regime
$sat(n,\mathcal{F}) \leq n+o(n)$. Finally, we essentially determine which
sequences of integers are of the form $\left(sat(n,\mathcal{F})\right)_{n \geq
0}$ for some (possibly infinite) family $\mathcal{F}$.
