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.
Categories: math.CO cs.DM