A graph is a minimal graph of diameter $2$ if it has diameter $2$ and the deletion of any edge increases its diameter.

Let $G$ be a minimal graph of diameter $2$, then every edge of $G$ belongs to a path $u$-$v$ in $G$ which is the unique path of length at most $2$ with endpoints $u$ and $v$.

My question is: If $H$ is a graph satisfies every edge of $H$ belongs to a path $u$-$v$ in $H$ which is the unique path of length at most $2$ with endpoints $u$ and $v$. Does there must exist a minimal graph $G$ of diameter $2$ such that $V(G)=V(H)$ and $H$ is a subgraph of $G$?