  It is known that the class of all graphs not containing a graph $H$ as an
induced subgraph is cop-bounded if and only if $H$ is a forest whose every
component is a path. In this study, we characterize all sets $\mathscr{H}$ of
graphs with some $k\in \mathbb{N}$ bounding the diameter of members of
$\mathscr{H}$ from above, such that $\mathscr{H}$-free graphs, i.e. graphs with
no member of $\mathscr{H}$ as an induced subgraph, are cop-bounded. This, in
particular, gives a characterization of cop-bounded classes of graphs defined
by a finite set of connected graphs as forbidden induced subgraphs.
Furthermore, we extend our characterization to the case of cop-bounded classes
of graphs defined by a set $\mathscr{H}$ of forbidden graphs such that there is
$k\in\mathbb{N}$ bounding the diameter of components of members of
$\mathscr{H}$ from above.
