  We introduce and study quantized versions of Cop and Robber game. We achieve
this by using graph-preserving quantum operations, which are the quantum
analogues of stochastic operations preserving the graph. We provide the tight
bound for the number of operations required to reach the given state. By
extending them to the controlled operations, we define a quantum-controlled Cop
and Robber game, which expands the classical Cop and Robber game, as well as
the classically controlled quantum Cop and Robber game. In contrast to the
typical scheme for introducing quantum games, we assume that both parties can
utilise full information about the opponent's strategy. We show that the
utilisation of the full knowledge about the opponent's state does not provide
the advantage. Moreover, the chances of catching the Robber decrease for
classical cop-win graphs. This result does not depend on the chosen model of
evolution. On the other hand, the possibility to execute controlled quantum
operations allows catching the Robber on almost all classical cop-win graphs.
By this, we demonstrate that it is necessary to enrich the structure of
correlations between the players' systems to provide a non-trivial quantized
Cop and Robber game. Thus the quantum controlled operations offer a significant
advantage over the classically controlled quantum operations.
