Cops and Robbers on Graphs with a Set of Forbidden Induced Subgraphs

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.