The Petersen graph is the smallest 3-cop-win graph

In the game of cops and robbers on a graph G = (V,E), k cops try to catch a robber. On the cop turn, each cop may move to a neighboring vertex or remain in place. On the robber’s turn, he moves similarly. The cops win if there is some time at which a cop is at the same vertex as the robber. Otherwise, the robber wins. The minimum number of cops required to catch the robber is called the cop number of G, and is denoted c(G). Let mk be the minimum order of a connected graph satisfying c(G) ≥ k. Recently, Baird and Bonato determined via computer search thatm3 = 10 and that this value is attained uniquely by the Petersen graph. Herein, we give a self-contained mathematical proof of this result. Along the way, we give some characterizations of graphs with c(G) > 2 and very high maximum degree.