Known property? Maximum radius of connected induced subgraph
The property $\Pi_r$, defined as containing exactly the graphs $G$ such that every induced subgraph $H$ of $G$ has diameter at most $r$, is the same as the class of graphs that do not contain a $P_{r+2}$ as induced subgraph, where the $P_{r+2}$ is the path on $r+2$ vertices. 

The equivalence can be seen as follows. The two degree-one vertices in the $P_{r+2}$ have a distance of $r+1$. Hence, if $G$ contains a subgraph isomorphic to the $P_{r+2}$, then $G$ does not fulfill $\Pi_r$. Conversely, if $G$ does not fulfill $\Pi_r$, then $G$ has an induced subgraph $H$ containing two vertices $u$ and $v$ such that the distance between $u$ and $v$ in $H$ is at least $r+1$. Take the vertex set $P$ of a shortest path between $u$ and $v$ in $H$. Since $P$ is a shortest path, $G[P]$ is an induced path, that is, it is isomorphic to some $P_\ell$. Since $u$ and $v$ have distance at least $r+1$ in $G[P]$, we have $\ell\ge r+2$. Hence $G$ contains $P_{r+2}$ as an induced subgraph.