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

## TL;DR

This paper characterizes graph classes defined by forbidden induced subgraphs that are cop-bounded, extending known results to sets with bounded diameter components, providing a comprehensive understanding of such classes.

## Contribution

It offers a complete characterization of cop-bounded graph classes defined by finite sets of forbidden induced subgraphs with bounded diameter.

## Key findings

- Characterization of cop-bounded classes with forbidden induced subgraphs.
- Extension to sets with bounded diameter components.
- Provides criteria for cop-boundedness based on forbidden subgraph sets.

## Abstract

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.