On the isometric embedding of arbitrary graphs into a graph of a given diameter possessing the metric continuation property

We prove that an arbitrary ordinary graph $G$ can be embedded as a generated subgraph into a graph $H$ of given diameter $d(H)=d\geqslant 2$ in which any two vertices lie on some diametral path. If the diameter $d(G)$ of $G$ is less than or equal to $d$, then the embedding can be achieved isometrically, that is, with preservation of the distances between the vertices in $G$.