  We investigate extremal graphs related to the game of Cops and Robbers. We
focus on graphs where a single cop can catch the robber; such graphs are called
cop-win. The capture time of a cop-win graph is the minimum number of moves the
cop needs to capture the robber. We consider graphs that are extremal with
respect to capture time, i.e. their capture time is as large as possible given
their order. We give a new characterization of the set of extremal graphs. For
our alternative approach we assign a rank to each vertex of a graph, and then
study which configurations of ranks are possible. We partially determine which
configurations are possible, enough to prove some further extremal results. We
leave a full classification as an open question.
