  Let $\mathcal{H}$ be a set of given connected graphs. A graph $G$ is said to
be $\mathcal{H}$-free if $G$ contains no $H$ as an induced subgraph for any
$H\in \mathcal{H}$. The graph $G$ is super-edge-connected if each minimum
edge-cut isolates a vertex in $G$. In this paper, except for some special
graphs, we characterize all forbidden subgraph sets $\mathcal{H}$ such that
every $\mathcal{H}$-free is super-edge-connected for $|\mathcal{H}|=1$ and $2$.
