A witness version of the Cops and Robber game

The games considered are mixtures of Searching and Cops and Robber. The cops have partial information provided via witnesses who report “sightings” of the robber. The witnesses are able to provide information about the robber’s position but not the direction in which he is moving. The robber has perfect information. In the case when sightings occur at regular intervals, we present a recognition theorem for graphs on which a single cop suffices to guarantee a win. In a special case, this recognition theorem provides a characterization.