  A graph $G$ is $H$-saturated if it contains no copy of $H$ as a subgraph but
the addition of any new edge to $G$ creates a copy of $H$. In this paper we are
interested in the function sat$_{t}(n,p)$, defined to be the minimum number of
edges that a $K_{p}$-saturated graph on $n$ vertices can have if it has minimum
degree at least $t$. We prove that sat$_{t}(n,p) = tn - O(1)$, where the limit
is taken as $n$ tends to infinity. This confirms a conjecture of Bollob\'as
when $p = 3$. We also present constructions for graphs that give new upper
bounds for sat$_{t}(n,p)$ and discuss an analogous problem for saturated
hypergraphs.
