  We investigate the game of cops and robber, played on a finite graph, between
one cop and one robber. If the cop can force a win on a graph, the graph is
called cop-win. We describe a procedure we call corner ranking, performed on a
graph, which assigns a positive integer or $\infty$ to each vertex. We give a
characterization of cop-win in terms of corner rank and also show that the
well-known characterization of cop-win via dismantling orderings follows from
our work. From the corner rank we can determine the capture time of a graph,
i.e. the number of turns the cop needs to win. We describe a class of optimal
cop strategies we call Lower Way strategies, and a class of optimal robber
strategies we call Higher Way strategies. Roughly speaking, in a Lower Way
strategy, the cop pushes the robber down to lower ranked vertices, while in a
Higher Way strategy, the robber moves to a highest rank vertex that is "safe."
While interesting in their own right, the strategies are themselves tools in
our proofs. We investigate various properties of the Lower Way strategies.
