Let G = (V, E) be a graph and k a positive integer such that k ≤ | V | . The k -token graph of G is the graph F k (G) having the set of all k -sets of V as vertex set, and such that two k -sets of V , say A and B , are adjacent if and only if the symmetric difference of A and B is an edge of G. In this work we study the cop number of token graphs of graphs in some classic families, such as paths, stars, and subdivided stars. We obtain the exact cop number for k -token graphs of paths and stars. We also obtain the exact cop number for subdivided stars when the number of branches is large enough. Probably more interesting than the aforementioned results, we introduce a variant of the Cops and Robbers game where R controls a team of robbers and C controls some teams of cops. A game of Cops and Robbers with this new variant played on a graph G is equivalent to a classic game of Cops and Robbers played on F k (G). This turns out to be very useful, as F k (G) is usually very large and complex when compared to G .

The game of Cops and Robbers is a well-known game played on graphs. In this paper, we consider the class of graphs of bounded diameter. We improve the strategy of cops and the previously used probabilistic method, which results in an improved upper bound for the cop number of graphs of bounded diameter. In particular, for graphs of diameter 4, we improve the upper bound from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{3}{5}+o(1)}$ and for diameter 3 from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{4}{7}+o(1)}$.