Let $G$ be a simple graph. Let $E^-(G)$ denote the set of (isomorphism classes) of subgraphs of $G$ that can be obtained by deleting a single edge of $G$. Similarly, let $E^+(G)$ be the set of (isomorphism classes) of simple graphs that can be obtained by adding a single edge to $G$. Is $G$ uniquely determined by $E^-(G)$ and $E^+(G)$? Thank you in advance!

Are there any other examples besides the two graphs on 4 vertices?