In graph theory, the game of cops and robbers is played on a finite, connected graph. The players take turns moving along edges as the cops try to capture the robber and the robber tries to evade capture forever. This game has received quite a bit of recent attention including several conjectures that have yet to be proven. In this paper, we restricted our attention to planar graphs in order to try to prove the conjecture that the dodecahedron graph is the smallest planar graph, in terms of vertices, that has cop number three. Along the way we discuss several other graphs with interesting properties connected with the cop number including a proof that the Tutte graph has cop number two
