Characterizing the forbidden pairs for graphs to be super-edge-connected

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 $2$.