On the minimum order of $k$-cop win graphs

Abstract

We consider the minimum order graphs with a given cop number. We prove that the minimum order of a connected graph with cop number 3 is 10, and show that the Petersen graph is the unique isomorphism type of graph with this property. We provide the results of a computational search on the cop number of all graphs up to and including order 10. A relationship is presented between the minimum order of graph with cop number $k$ and Meyniel's conjecture on the asymptotic maximum value of the cop number of a connected graph.