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In the game of \emph{cops and robbers} on a graph $G = (V,E)$, $k$ cops tryto catch a robber. On the cop turn, each cop may move to a neighboring vertexor remain in place. On the robber's turn, he moves similarly. The cops win ifthere 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 therobber is called the \emph{cop number} of $G$, and is denoted $c(G)$. Let $m_k$be the minimum order of a connected graph satisfying $c(G) \geq k$. Recently,Baird and Bonato determined via computer search that $m_3=10$ and that thisvalue is attained uniquely by the Petersen graph. Herein, we give aself-contained mathematical proof of this result. Along the way, we give somecharacterizations of graphs with $c(G) >2$ and very high maximum degree.

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